Derivations with Values in the Ideal of $\tau$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

Abstract

Let ${{\mathcal {M}}}$ be a semifinite von Neumann algebra with a faithful normal semifinite trace $\tau$ and let ${{\mathcal {A}}}$ be an arbitrary von Neumann subalgebra of ${{\mathcal {M}}}$. We characterize the class of symmetric ideals ${{\mathcal {E}}}$ in ${{\mathcal {M}}}$ such that derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$ are necessarily inner, which is a unification and far-reaching extension of the results due to Johnson and Parrott (J Funct Anal 11:39–61, 1972), due to Kaftal and Weiss (J Funct Anal 62:202–220, 1985), and due to Popa (J Funct Anal 71:393–408, 1987). In particular, we show that every derivation from ${{\mathcal {A}}}$ into the ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ of all $\tau$-compact operators is inner, establishing a semifinite version of the Johnson–Parrott–Popa Theorem which is different from Popa and Rădulescu (Duke Math J 57(2):485–518, 1988, Theorem 1.1) and contrasts to the example of a non-inner derivation established in Popa and Rădulescu (1988, Theorem 1.2).

1 Introduction

Let ${{\mathcal {A}}}$ be a von Neumann algebra and let J be an ${{\mathcal {A}}}$-bimodule. A derivation $\delta :{{\mathcal {A}}}\rightarrow J$ is a linear mapping satisfying $\delta (XY) = \delta (X)Y + X\delta (Y)$, $X,Y \in {{\mathcal {A}}}$. In particular, if $K\in J$, then $\delta _K(X): = XK-KX$ is a derivation. Such derivations implemented by elements in J are called inner. One of the classical problems in operator algebra theory is the question whether every derivation from ${{\mathcal {A}}}$ into J is automatically inner. During the past decades, a number of important special cases have been resolved (see e.g. [4, 36, 37, 48, 54, 57]).

Kadison [38] and Sakai [58] gave an affirmative answer to the special case when the ${{\mathcal {A}}}$-bimodule J coincides with the algebra ${{\mathcal {A}}}$ itself. Further, it was proved that every derivation from a von Neumann algebra into its arbitrary ideal is automatically inner [12, 13]. However, when one considers more general ${{\mathcal {A}}}$-bimodules J (see e.g. [1, 2, 4,5,6,7,8,9,10,11, 35, 44, 55]), there are examples of non-inner derivations for some specific ${{\mathcal {A}}}$ and J (see e.g. [55], see also [7, 10, 59]). In the present paper, we concentrate on derivations with values in an ideal of ${{\mathcal {M}}}$, where ${{\mathcal {M}}}$ is an algebra such that ${{\mathcal {A}}}\subset {{\mathcal {M}}}$.

Question 1.1

Assume that ${{\mathcal {M}}}$ is a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Let ${{\mathcal {E}}}$ be an ideal in ${{\mathcal {M}}}$ and let ${{\mathcal {A}}}$ be an arbitrary von Neumann subalgebra of ${{\mathcal {M}}}$. Is every derivation $\delta$ from ${{\mathcal {A}}}$ into ${{\mathcal {E}}}$ necessarily inner?

It is desirable to identify the ideals ${{\mathcal {E}}}$ in ${{\mathcal {M}}}$ such that every derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$ is necessarily inner (see e.g. [33] and [36, Section 10.11]) and a number of mathematicians have been working on this problem since the 1960s (see e.g. [10, 12, 13, 16, 32, 33, 37, 38, 44, 54, 58]). Even though the special case when ${{\mathcal {E}}}={{\mathcal {M}}}$ is still open (e.g. when ${{\mathcal {M}}}=B({{\mathcal {H}}})$, the algebra of all bounded linear operators on a Hilbert space ${{\mathcal {H}}}$), there are results giving affirmative answers to this question under some additional conditions on the subalgebra ${{\mathcal {A}}}$ [53, 61]. For those who are interested in this special case, we refer to [15, 17, 18, 21, 53, 61].

Johnson and Parrott [37] considered the special case when the range of $\delta$ is contained in $K({{\mathcal {H}}})$, the ideal of all compact operators on ${{\mathcal {H}}}$, and $\delta$ acts on a von Neumman subalgebra ${{\mathcal {A}}}$ of $B({{\mathcal {H}}})$ having no certain type $II_1$ factors as direct summands. The remaining case, when ${{\mathcal {A}}}$ is a type $II_1$ von Neumann algebra was later resolved by Popa in [54].

Over the past decades, many authors have tried to establish a suitable semifinite version of the Johnson–Parrott–Popa Theorem [37, 54]. In [44], the derivation problem initiated by Johnson and Parrott was studied in a more general setting where $B({{\mathcal {H}}})$ is replaced with a semifinite von Neumann algebra ${{\mathcal {M}}}$ and $K({{\mathcal {H}}})$ is replaced with the uniform norm closed ideal ${{\mathcal {J}}}({{\mathcal {M}}})$ generated by all finite projections in ${{\mathcal {M}}}$. It is shown in [44] that if ${{\mathcal {A}}}$ is an abelian (or properly infinite) von Neumann subalgebra of ${{\mathcal {M}}}$ containing the center ${{\mathcal {Z}}}({{\mathcal {M}}})$ of ${{\mathcal {M}}}$, then any derivation of ${{\mathcal {A}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ is inner. Later, Popa and Rădulescu [55, Theorem 1.1] extended this result to the case when ${{\mathcal {A}}}$ is a von Neumann subalgebra of ${{\mathcal {M}}}$ and the finite type I summand of ${{\mathcal {A}}}$ is locally compatible with the center of ${{\mathcal {M}}}$ (in particular, they resolved the case when ${{\mathcal {A}}}$ is a type $II_1$ von Neumann subalgebra of ${{\mathcal {M}}}$, which was left untreated in [44]). Surprisingly, they established the existence of non-inner derivations $\delta : {{\mathcal {A}}}\rightarrow {{\mathcal {J}}}({{\mathcal {M}}})$ for some specific semifinite von Neumann algebra ${{\mathcal {M}}}$ and abelian von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$. The notion of the so-called generalized compacts $C({{\mathcal {M}}})$ associated with a von Neumann algebra ${{\mathcal {M}}}$ was introduced by Wright [66] in 1984. Christensen [16] showed that derivations $\delta : {{\mathcal {M}}}\rightarrow C({{\mathcal {M}}})$ are inner if ${{\mathcal {M}}}$ is properly infinite or injective. However, the case when ${{\mathcal {M}}}$ is a type $II_1$ von Neumann algebra was a long-standing open question and was resolved by Galatan and Popa [32] in 2017.

However, for a general semifinite von Neumann algebra ${{\mathcal {M}}}$, there is another notion of compactness, so-called $\tau$-compactness, which comes from a semifinite trace $\tau$ defined on the algebra ${{\mathcal {M}}}$. Hence, it is natural to consider a problem, similar to that considered in [55], with the ideal ${{\mathcal {J}}}({{\mathcal {M}}})$ of all compact operators replaced by the ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ of all $\tau$-compact operators affiliated with ${{\mathcal {M}}}$ (see Sect. 2 for precise definitions). The following theorem is first main result of our paper.

Theorem 1.2

Assume that ${{\mathcal {M}}}$ is a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Every derivation from an arbitrary von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner.

Although, the ideals ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and ${{\mathcal {J}}}({{\mathcal {M}}})$ are quite similar in many respects (see Sect. 2), the result of Theorem 1.2 is in strong contrast with [55, Theorem 1.1], which is somewhat unexpected. Namely, the additional assumption on the type I summand of the von Neumann subalgebra ${{\mathcal {A}}}$ which plays an important role in [55] could be dispensed with in our current setting. We note that the assumption that ${{\mathcal {A}}}$ is von Neumann subalgebra of ${{\mathcal {M}}}$ is sharp, that is, this assumption can not be relaxed to the assumption that ${{\mathcal {A}}}$ is a $C^*$-subalgebra of ${{\mathcal {M}}}$ for derivations into ${{\mathcal {J}}}({{\mathcal {M}}})$ or into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ (see [59, Example 4.1.8] and [10, Theorem 3.8]).

Derivations with values in ideals of a von Neumann algebra have important applications in the study of hyperreflexivity of operator spaces [50], commutants mod normed ideals [65], automorphisms and epimorphisms of operator algebras [12, 13, 39, 40], etc. In Sect. 8, as an application of Theorem 1.2, we show that Question 1.1 has a positive answer for a very extensive class of symmetric ideals ${{\mathcal {E}}}$ of ${{\mathcal {M}}}$ by using techniques concerning noncommutative integration, which are completely different from those used in [10, 33, 37, 44, 54, 55].

Theorem 1.3

Assume that ${{\mathcal {M}}}$ is a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. If $E({{\mathcal {M}}},\tau )$ is a strongly symmetric space of $\tau$-measurable operators with the Fatou property, then every derivation $\delta$ from an arbitrary von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into $E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is necessarily inner.

In [44], Kaftal and Weiss showed that derivations from ${{\mathcal {A}}}$ into ${{\mathcal {C}}}_p({{\mathcal {M}}},\tau ):=L_p({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$, $p\ge 1$, are necessarily inner if ${{\mathcal {A}}}$ is an abelian (or properly infinite) von Neumann subalgebra of ${{\mathcal {M}}}$, where $L_p({{\mathcal {M}}},\tau )$ stands for the noncommutative $L_p$-space. However, the case of type I (or type $II_1$) von Neumann subalgebras was left unresolved in [44]. Theorem 1.3 generalizes the results due to Kaftal and Weiss [44] in two directions. Firstly, any additional conditions imposed on the von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ in [44] are removed. In particular, we completely resolve the cases unanswered in [44, Section 6]. Secondly, we have extended significantly the class of symmetric ideals associated with ${{\mathcal {M}}}$ for which the result is applicable. One should note that the Fatou property is an analogue of the so-called “dual normal” property (see e.g. [61]) and it is well-known that every derivation from a hyperfinite von Neumann algebra ${{\mathcal {A}}}$ into a dual normal ${{\mathcal {A}}}$-bimodule is inner (see e.g. [61, Theorem 2.4.3]). However, Theorem 1.3 is applicable for any von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$. Moreover, this result complements Theorem 3.5 in our recent paper [10], where the derivations act on a larger class of algebras (i.e. $C^*$-subalgebras of ${{\mathcal {M}}}$) but the result is only applicable for symmetric operator spaces $E({{\mathcal {M}}},\tau )$ corresponding to Kantorovich–Banach fully symmetric function spaces $E(0,\infty )$ [24], that is, the additional assumption in [10] that the norm of $E(0,\infty )$ is order continuous is omitted in Theorem 1.3.

2 Preliminaries

In this section, we recall some notions of the theory of noncommutative integration.

In what follows, ${{\mathcal {H}}}$ is a Hilbert space and $B({{\mathcal {H}}})$ is the $*$-algebra of all bounded linear operators on ${{\mathcal {H}}}$ equipped with the uniform norm $\left\| \cdot \right\| _\infty$, and ${\mathbf {1}}$ is the identity operator on ${{\mathcal {H}}}$. Let ${\mathcal {M}}$ be a von Neumann algebra on ${{\mathcal {H}}}$. We denote by ${{\mathcal {P}}}({{\mathcal {M}}})$ the collection of all projections in ${{\mathcal {M}}}$, by ${{\mathcal {M}}}'$ the commutant of ${{\mathcal {M}}}$ and by ${{\mathcal {Z}}}({{\mathcal {M}}})$ the center of ${{\mathcal {M}}}$. For details on von Neumann algebra theory, the reader is referred to e.g. [22, 41, 42] or [62]. General facts concerning measurable operators may be found in [49, 60] (see also [63, Chapter IX] and the forthcoming book [30]). For convenience of the reader, some of the basic definitions are recalled.

2.1 $\tau$-measurable operators and generalized singular value functions

A closed, densely defined operator $X:{\mathfrak {D}}\left( X\right) \rightarrow {{\mathcal {H}}}$ with the domain ${\mathfrak {D}}\left( X\right)$ is said to be affiliated with ${\mathcal {M}}$ if $YX\subseteq XY$ for all $Y\in {\mathcal {M}}^{\prime }$, where ${\mathcal {M}}^{\prime }$ is the commutant of ${\mathcal {M}}$. A closed, densely defined operator $X:{\mathfrak {D}}\left( X\right) \rightarrow {{\mathcal {H}}}$ affiliated with ${{\mathcal {M}}}$ is said to be measurable if there exists a sequence $\left\{ P_n\right\} _{n=1}^{\infty }\subset {{\mathcal {P}}}\left( {\mathcal {M}}\right)$, such that $P_n\uparrow {\mathbf {1}}$, $P_n({{\mathcal {H}}})\subseteq {\mathfrak {D}}\left( X\right)$ and ${\mathbf {1}}-P_n$ is a finite projection (with respect to ${\mathcal {M}}$) for all n. The collection of all measurable operators with respect to ${\mathcal {M}}$ is denoted by $S\left( {\mathcal {M}} \right)$, which is a unital $*$-algebra with respect to strong sums and products (denoted simply by $X+Y$ and XY for all $X,Y\in S\left( \mathcal {M }\right)$).

Let X be a self-adjoint operator affiliated with ${\mathcal {M}}$. We denote its spectral measure by $\{E^X\}$. It is well known that if X is an operator affiliated with ${\mathcal {M}}$ with the polar decomposition $X = U|X|$, then $U\in {\mathcal {M}}$ and $E\in {\mathcal {M}}$ for all projections $E\in \{E^{|X|}\}$. Moreover, $X\in S({\mathcal {M}})$ if and only if $E^{|X|}(\lambda , \infty )$ is a finite projection for some $\lambda > 0$. It follows immediately that in the case when ${\mathcal {M}}$ is a von Neumann algebra of type III or a type I factor, we have $S({\mathcal {M}})= {\mathcal {M}}$. For type II von Neumann algebras, this is no longer true. From now on, let ${\mathcal {M}}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$.

An operator $X\in S\left( {\mathcal {M}}\right)$ is called $\tau$-measurable if there exists a sequence $\left\{ P_n\right\} _{n=1}^{\infty }$ in $P\left( {\mathcal {M}}\right)$ such that $P_n\uparrow {\mathbf {1}}$, $P_n\left( {{\mathcal {H}}}\right) \subseteq {\mathfrak {D}}\left( X\right)$ and $\tau ({\mathbf {1}}-P_n)<\infty$ for all n. The collection $S\left( {\mathcal {M}}, \tau \right)$ of all $\tau$-measurable operators is a unital $*$-subalgebra of $S\left( {\mathcal {M}}\right)$. It is well known that a linear operator X belongs to $S\left( {\mathcal {M}}, \tau \right)$ if and only if $X\in S({\mathcal {M}})$ and there exists $\lambda >0$ such that $\tau (E^{|X|}(\lambda , \infty ))<\infty$. Alternatively, an unbounded operator X affiliated with ${\mathcal {M}}$ is $\tau$-measurable (see [31]) if and only if

$$\begin{aligned} \tau \left( E^{|X|}\bigl (n,\infty \bigr )\right) \rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

Definition 2.1

Let a semifinite von Neumann algebra ${\mathcal {M}}$ be equipped with a faithful normal semi-finite trace $\tau$ and let $X\in S({\mathcal {M}},\tau )$. The generalized singular value function $\mu (X):t\rightarrow \mu (t;X)$ of the operator X is defined by setting

$$\begin{aligned} \mu (s;X) = \inf \{\Vert XP\Vert :\ P\in {{\mathcal {P}}}({{\mathcal {M}}}) \text{ with }\ \tau ({\mathbf {1}}-P)\le s\}. \end{aligned}$$

An equivalent definition in terms of the distribution function of the operator X is the following. For every self-adjoint operator $X\in S({\mathcal {M}},\tau ),$ setting

$$\begin{aligned} d_X(t)=\tau (E^{X}(t,\infty )),\quad t>0, \end{aligned}$$

we have (see e.g. [31])

$$\begin{aligned} \mu (t; X)=\inf \{s\ge 0:\ d_{|X|}(s)\le t\}. \end{aligned}$$

(2.1)

It is well-known [28, 47] that if $A\in S({{\mathcal {M}}},\tau )$ and $B,C\in {{\mathcal {M}}}$, then

$$\begin{aligned} \mu (t;BAC)\le \left\| B\right\| _\infty \left\| C\right\| _\infty \mu (t;A), ~\mu (t;A^*)=\mu (t;A). \end{aligned}$$

(2.2)

Consider the algebra ${\mathcal {M}}=L^\infty (0,\infty )$ of all Lebesgue measurable essentially bounded functions on $(0,\infty )$. The algebra ${\mathcal {M}}$ can be seen as an abelian von Neumann algebra acting via multiplication on the Hilbert space ${\mathcal {H}}=L^2(0,\infty )$, with the trace given by integration with respect to Lebesgue measure m. It is easy to see that the algebra of all $\tau$-measurable operators affiliated with ${\mathcal {M}}$ can be identified with the subalgebra $S(0,\infty )$ of the algebra of Lebesgue measurable functions $L_0(0,\infty )$ which consists of all functions x such that $m(\{|x|>s\})$ is finite for some $s>0$. It should also be pointed out that the generalized singular value function $\mu (x)$ is precisely the decreasing rearrangement $\mu (x)$ of the function |x| (see e.g. [46]) defined by

$$\begin{aligned} \mu (t;x)=\inf \{s\ge 0:\ m(\{|x|\ge s\})\le t\}. \end{aligned}$$

If ${\mathcal {M}}=B({{\mathcal {H}}})$ (respectively, $l_\infty$) and $\tau$ is the standard trace $\mathrm{Tr}$ (respectively, the counting measure on ${\mathbb {N}}$), then it is not difficult to see that $S({\mathcal {M}})=S({\mathcal {M}},\tau )={\mathcal {M}}.$ In this case, for $X\in S({\mathcal {M}},\tau )$ we have

$$\begin{aligned} \mu (n;X)=\mu (t; X),\quad t\in [n,n+1),\quad n\ge 0. \end{aligned}$$

The sequence $\{\mu (n;X)\}_{n\ge 0}$ is just the sequence of singular values of the operator X.

If $X,Y\in S({{\mathcal {M}}},\tau )$, then X is said to be submajorized by Y, denoted by $X\prec \prec Y$, if

$$\begin{aligned} \int _{0}^{t} \mu (s;X) ds \le \int _{0}^{t} \mu (s;Y) ds \end{aligned}$$

for all $t\ge 0$. In particular, for $x,y\in S(0,\infty )$, $x\prec \prec y$ if and only if $\int _{0}^{t} \mu (s;x) ds \le \int _{0}^{t} \mu (s;y) ds$, $t\ge 0$.

The following proposition is an easy consequence of [29, Lemma 6.1] and [47, Lemma 3.3.5].

Proposition 2.2

Assume that $P_1,P_2,\cdots , P_n\in {{\mathcal {M}}}$ are projections with $P_iP_j =0$, $i\ne j$. Let $\alpha _i>0$, $i\in {\mathbb {N}}$, be such that $\sum _{i=1}^n \alpha _i \le 1$. For every $X\in S({{\mathcal {M}}},\tau )$, we have

$$\begin{aligned} \int _0^a \mu (t;X )dt \ge \sum _{i=1}^n \int _0^{\alpha _i a} \mu (t; | P_iX P_i| )dt= \sum _{i=1}^n \int _0^{\alpha _i a} \mu (t; P_i X P_i )dt, ~\forall a>0. \end{aligned}$$

(2.3)

Proof

Since $P_i$ are pairwise disjoint, it follows that

$$\begin{aligned} \mu (P_1XP_1 +P_2XP_2+\cdots +P_nXP_n)&=\mu (|P_1XP_1 +P_2XP_2+\cdots +P_nXP_n|)\\&=\mu (|P_1XP_1| +|P_2XP_2|+\cdots +|P_nXP_n|). \end{aligned}$$

Therefore, by [29, Lemma 6.1], we obtain that

$$\begin{aligned} \int _0^a \mu (t;|P_1XP_1| +|P_2XP_2|+\cdots +|P_nXP_n| )dt \le \int _0^a \mu (t;X )dt. \end{aligned}$$

The validity of (2.3) follows from [47, Lemma 3.3.5]. $\quad \square$

The following lemmas provide useful tools in proving the main theorem of the present paper.

Lemma 2.3

If $\{T_i \}\subset {{\mathcal {M}}}$ is a uniformly bounded net of self-adjoint operators converging to $T\in {{\mathcal {M}}}$ in the strong operator topology, then

$$\begin{aligned} \liminf _i \tau (E^{T_i}(\varepsilon ,\infty )) \ge \tau (E^{T}(\varepsilon ,\infty )) \end{aligned}$$

for any $\varepsilon \in {\mathbb {R}}$.

Proof

Consider the characteristic function $\chi _{(\varepsilon ,\infty )}$. There exists a sequence of positive continuous functions $f_k$ with compact support such that $f_k\uparrow \chi _{(\varepsilon ,\infty )}$ pointwise. By [62, Lemma II 4.3], we have $f_k(T_i)\rightarrow _{so} f_k(T)$ for all $k\in {\mathbb {N}}$. Since $\tau$ is lower semicontinuous in the weak operator topology on a uniformly bounded set (see e.g. [62, Lemma II 2.5] and [63, Theorem VII 1.11]), it follows that

$$\begin{aligned} \tau (f_k(T))\le \liminf _{i}\tau (f_k(T_i))\le \liminf _i \tau (\chi _{(\varepsilon ,\infty )}(T_i))=\liminf _i \tau (E^{T_i}(\varepsilon ,\infty )). \end{aligned}$$

Note that $f_k\uparrow \chi _{(\varepsilon ,\infty )}$ implies $\sup _k f_k(T)=\chi _{(\varepsilon ,\infty )}(T)=E^{T}(\varepsilon ,\infty )$. Hence, using the normality of the trace $\tau$, we conclude that

$$\begin{aligned} \tau (E^{T}(\varepsilon ,\infty ))=\sup _{k\in {\mathbb {N}}}\tau (f_k(T))\le \liminf _i \tau (E^{T_i}(\varepsilon ,\infty )). \end{aligned}$$

$\square$

Lemma 2.4

For every $X\in {{\mathcal {M}}}$ and $t>0$, $\tau (E^{|X|}(a,\infty )) > t$ if and only if $\mu (s; X) > a$ for all $s\in [0,t]$.

Proof

Necessity. By (2.1), we have $\mu (t; X) = \inf \{s \ge 0 : \tau (E^{|X|}(s,\infty ))\le t \}$. Assume by contradiction that $\mu (t;X)\le a$, then $\inf \{s \ge 0 : \tau (E^{|X|}(s,\infty ))\le t \} \le a$ and therefore $\tau (E^{|X|}(a+\varepsilon ,\infty ))\le t$ for any $\varepsilon >0$. Since the distribution function $d_{|X|}(\cdot )$ is right-continuous (see e.g. [31]), it follows that $\tau (E^{|X|}(a,\infty )) \le t$, which is a contradiction.

Sufficiency. By assumption, we have that $\mu (t; X) > a$. Using again (2.1), we obtain that $\inf \{s \ge 0 : \tau (E^{|X|}(s,\infty ))\le t \} > a$, and therefore $\tau (E^{|X|}(a,\infty ))> t$. $\quad \square$

2.2 Symmetric spaces

Definition 2.5

A linear subspace E of $S({{\mathcal {M}}},\tau )$ equipped with a complete norm $\Vert \cdot \Vert _E$, is called symmetric space (of $\tau$-measurable operators) if $X\in S({{\mathcal {M}}},\tau )$, $Y \in E$ and $\mu (X)\le \mu (Y)$ imply that $X\in E$ and $\left\| X\right\| _E \le \left\| Y\right\| _E$.

It is well-known that any symmetric space E is a normed ${{\mathcal {M}}}$-bimodule, that is $AXB\in E$ for any $X\in E$, $A,B\in {{\mathcal {M}}}$ and $\left\| AXB\right\| _E\le \Vert A\Vert _\infty \left\| B\right\| _\infty \left\| X\right\| _E$ [28, 30].

A symmetric space $E({{\mathcal {M}}},\tau )\subset S({{\mathcal {M}}},\tau )$ is called strongly symmetric if its norm $\Vert \cdot \Vert _E$ has the additional property that $\Vert X\Vert _E \le \Vert Y\Vert _E$ whenever $X,Y \in E({{\mathcal {M}}},\tau )$ satisfy $X\prec \prec Y$. In addition, if $X\in S({{\mathcal {M}}},\tau )$, $Y \in E({{\mathcal {M}}},\tau )$ and $X\prec \prec Y$ imply that $X\in E({{\mathcal {M}}},\tau )$ and $\Vert X\Vert _E \le \Vert Y\Vert _E$, then $E({{\mathcal {M}}},\tau )$ is called fully symmetric space (of $\tau$-measurable operators).

A symmetric space $E({{\mathcal {M}}},\tau )$ is said to have the Fatou property if for every upwards directed net $\{X_\beta \}$ in $E({{\mathcal {M}}},\tau )^+$, satisfying $\sup _\beta \left\| X_\beta \right\| _E <\infty$, there exists an element $X\in E({{\mathcal {M}}},\tau )^+$ such that $X_\beta \uparrow X$ in $E({{\mathcal {M}}},\tau )$ and $\left\| X\right\| _E = \sup _\beta \left\| X_\beta \right\| _E$. Examples such as Schatten-von Neumann operator ideals, Lorentz operator ideals, Orlicz operator ideals, etc. all have symmetric norms which have the Fatou property.

If $E({{\mathcal {M}}},\tau )$ is a symmetric space, then the carrier projection $c_E\in {{\mathcal {P}}}({{\mathcal {M}}})$ is defined by setting

$$\begin{aligned} c_E = \bigvee \{P:P\in P({{\mathcal {M}}}),~P \in E({{\mathcal {M}}},\tau )\}. \end{aligned}$$

If $E ({{\mathcal {M}}},\tau )$ is a symmetric space, then the Köthe dual $E({{\mathcal {M}}},\tau )^\times$ of $E({{\mathcal {M}}},\tau )$ is defined by

$$\begin{aligned} E({{\mathcal {M}}},\tau )^\times =\{ X\in S({{\mathcal {M}}},\tau ) : \sup _{\Vert Y\Vert _E\le 1, Y\in E}\tau (|XY|) <\infty \}, \end{aligned}$$

and for every $X\in E({{\mathcal {M}}},\tau )^\times$, we set $\left\| X\right\| _{E^\times } = \sup \{\tau (|YX|) : ~Y\in E({{\mathcal {M}}},\tau ), ~\left\| Y\right\| _E \le 1\}$ (see e.g. [28, Section 5.2], see also [25, 47]). It is well-known that $\left\| \cdot \right\| _{E^\times }$ is a norm on $E({{\mathcal {M}}},\tau )^\times$ if and only if the carrier projection $c_E$ of $E({{\mathcal {M}}},\tau )$ is equal to $\mathbf{1}$. In this case, for a strongly symmetric space $E({{\mathcal {M}}},\tau )$, the following statements are equivalent [26, 28].

• $E({{\mathcal {M}}},\tau )$ has the Fatou property.

• $E({{\mathcal {M}}},\tau )^{\times \times }=E({{\mathcal {M}}},\tau )$ and $\left\| X\right\| _E=\left\| X\right\| _{E^{\times \times }}$ for all $X\in E({{\mathcal {M}}},\tau )$.

A wide class of symmetric operator spaces associated with the von Neumman algebra ${{\mathcal {M}}}$ can be constructed from concrete symmetric function spaces studied extensively in e.g. [46]. Let $(E(0,\infty ),\left\| \cdot \right\| _{E(0 ,\infty )})$ be a symmetric function space on the semi-axis $(0,\infty )$. The pair

$$\begin{aligned} E({{\mathcal {M}}},\tau )=\{X\in S({{\mathcal {M}}},\tau ):\mu (X)\in E(0,\infty )\},\quad \Vert X\Vert _{E({{\mathcal {M}}},\tau )}:=\Vert \mu (X)\Vert _{E(0,\infty )} \end{aligned}$$

is a symmetric operator space affiliated with ${{\mathcal {M}}}$ with $c_E =\mathbf{1}$ [45] (see also [47]). Further, we have

$$\begin{aligned} L_1\cap L_\infty ({{\mathcal {M}}},\tau )\subset E({{\mathcal {M}}},\tau ) \subset (L_1+ L_\infty ) ({{\mathcal {M}}},\tau ). \end{aligned}$$

For convenience, we denote $\left\| \cdot \right\| _{E({{\mathcal {M}}},\tau )}$ by $\left\| \cdot \right\| _E$. Many properties of symmetric spaces, such as reflexivity, Fatou property, order continuity of the norm as well as Köthe duality carry over from commutative symmetric function space $E(0,\infty )$ to its noncommutative counterpart $E({{\mathcal {M}}},\tau )$ (see e.g. [28, Theorem 53 and Theorem 54] and [26]).

2.3 The ideal of $\tau$-compact operators

A projection $P\in {{\mathcal {P}}}({{\mathcal {M}}})$ is called $\tau$-finite if $\tau (P)<\infty$. If $P\in {{\mathcal {P}}}({{\mathcal {M}}})$ is $\tau$-finite, then P is a finite projection. For $X\in {{\mathcal {M}}}$, we denote by l(X) and r(X) the left support and right support of X, respectively. In particular, for a self-adjoint operator $X\in {{\mathcal {M}}}$, we denote by s(X) its support. Every projection $P\in {{\mathcal {M}}}$ has a central support which is defined as the smallest projection in the center ${{\mathcal {Z}}}({{\mathcal {M}}})$ containing P as a subprojection. The two-sided ideal ${{\mathcal {F}}}({{\mathcal {M}}},\tau )$ in ${{\mathcal {M}}}$ consisting of all elements of $\tau$-finite range is defined by setting

$$\begin{aligned} {{\mathcal {F}}}({{\mathcal {M}}},\tau )=\{X\in {{\mathcal {M}}}~:~ \tau (r(X))<\infty \} = \{X\in {{\mathcal {M}}}~:~ \tau (l(X)) <\infty \}. \end{aligned}$$

The ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ of all $\tau$-compact bounded operators can be described as the closure in the norm $\Vert \cdot \Vert _\infty$ of the linear span of all $\tau$-finite projections [47, Definition 2.6.8]. Equivalently, ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is set of all elements $X\in {{\mathcal {M}}}$ such that $\tau (E^{|X|}(\lambda ,\infty ))<\infty$ for every $\lambda >0$ (see e.g. [30, Chapter II, Section 4]). The space ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is associated to the ideal of essentially bounded functions vanishing at infinity (see [47, Lemma 2.6.9]), that is,

$$\begin{aligned} {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) = \{ A \in S({{\mathcal {M}}},\tau ) : \mu (A)\in L_\infty (0,\infty ), \ \mu (\infty ;A):=\lim _{t\rightarrow \infty }\mu (t;A) =0\}. \end{aligned}$$

In particular, if $\tau$ is finite, then ${{\mathcal {M}}}= {{\mathcal {C}}}_0( {{\mathcal {M}}},\tau )$ (see e.g. [47, Page 64]). The space $S_0({{\mathcal {M}}},\tau )$ of $\tau$-compact operators is the space associated to the algebra of functions from $S(0,\infty )$ vanishing at infinity, that is,

$$\begin{aligned} S_0({{\mathcal {M}}},\tau ) = \{A\in S({{\mathcal {M}}},\tau ) : \ \mu (\infty ; A) =0\}. \end{aligned}$$

This is a two-sided ideal in $S({{\mathcal {M}}},\tau )$ and, clearly, ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) = S_0({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$.

The following lemma provides a sufficient condition for an operator $X\in {{\mathcal {M}}}$ to be not $\tau$-compact. This condition plays a crucial role in the proof of Theorem 4.2.

Lemma 2.6

Let $X\in {{\mathcal {M}}}$ and $\{\alpha _i>0\}_i$ be an arbitrary sequence of real numbers increasing to infinity. If there exists a number $c>0$ such that

$$\begin{aligned} \int _{0}^{\alpha _i} \mu (t;X)dt \ge \alpha _i c \end{aligned}$$

for every $\alpha _i$, then $\mu (t;X) \ge c$ for all $t>0$, that is, X is not $\tau$-compact. In other words, if $\mu (X) \succ \succ c$, then $\mu (X)\ge c$.

Proof

Assume by contradiction that $\mu (n_0;X) <c$ for some $n_0 >0$ and therefore $\mu (t; X) \le \mu (n_0;X) < c$ for every $t\ge n_0$. Noticing that $\Vert X\Vert _\infty <\infty$, we have

$$\begin{aligned} n_0 \cdot \Vert X\Vert _\infty + (\alpha _i -n_0) \mu (n_0;X)&\ge \int _0^{n_0} \Vert X\Vert _\infty dt + \int _{n_0}^{\alpha _i} \mu (n_0;X) dt \\&\ge \int _0^{n_0}\mu (t; X)dt + \int _{n_0}^{\alpha _i}\mu (t; X)dt \\&= \int _0^{\alpha _i} \mu (t; X)dt \ge \alpha _i c \end{aligned}$$

for any $\alpha _i\ge n_0$. Thus, we obtain that $\mu (n_0;X) \ge \frac{\alpha _i c -n_0\Vert X\Vert _\infty }{\alpha _i-n_0}$ for every $\alpha _i\ge n_0$. By assumption, we have that $\alpha _i\rightarrow _i \infty$ as $i\rightarrow \infty$, and therefore, it follows that $\mu (n_0;X)\ge c$, which is a contradiction. Thus, $\mu (t;X)\ge c$ for all $t>0$, which implies that the operator X is not $\tau$-compact. $\quad \square$

Recall that ${{\mathcal {J}}}({{\mathcal {M}}})$ is the uniform norm closure of the linear span of all finite projections in ${{\mathcal {M}}}$. It is known that ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {J}}}({{\mathcal {M}}})$ for any semifinite algebra ${{\mathcal {M}}}$ and ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )= {{\mathcal {J}}}({{\mathcal {M}}})$ whenever ${{\mathcal {M}}}$ is a factor (see e.g. [55, 2.1.1.]).

Remark 2.7

For a semifinite von Neumann algebra ${\mathcal {M}}$ equipped with a faithful normal semifinite trace $\tau$, it is easy to see that ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\ne {{\mathcal {J}}}({{\mathcal {M}}})$ if and only if there exists a finite projection $P\in {{\mathcal {M}}}$ such that $\tau (P)=\infty$ (see e.g. [43, Theorem 1.3]).

We end this section with the following theorem, which gives a necessary and sufficient condition on the algebra ${{\mathcal {M}}}$ for the existence of a faithful normal semifinite trace $\tau$ on ${{\mathcal {M}}}$ with ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subsetneqq {{\mathcal {J}}}({{\mathcal {M}}})$.

Theorem 2.8

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra. The following conditions are equivalent:

• (i). There exists a faithful normal semifinite trace $\tau$ on ${\mathcal {M}}$ such that ${{\mathcal {C}}}_0({\mathcal {M}},\tau )\ne {{\mathcal {J}}}({\mathcal {M}})$;

• (ii). $\dim ({\mathcal {Z}}({\mathcal {M}}))=\infty$.

Proof

(i)$\Rightarrow$(ii). Assume by contradiction that $\dim ({\mathcal {Z}}({\mathcal {M}}))<\infty$. We denote by $E_1,\dots ,E_n$, $n\in {\mathbb {N}}$, the finite family of atoms in ${{\mathcal {Z}}}({{\mathcal {M}}})$. It is clear that ${\mathcal {M}}_{E_k}$ is a semifinite factor for all $k=1,\dots ,n$. For every $k=1,\dots ,n$, fix a trace $\tau _k$ on ${\mathcal {M}}_{E_k}$. It is clear that $\tau (X)=\sum _{k=1}^n \alpha _k\tau _k(XE_k)$ for some $\alpha _k>0$.

By Remark 2.7, we can find a finite projection $P\in {{\mathcal {P}}}({{\mathcal {M}}})$ such that $\tau (P)=\infty$. Therefore, $\tau _k(PE _k)=\infty$ for some k. However, this is impossible since $PE_k$ is a finite projection in the factor ${\mathcal {M}}_{E_k}$. This contradiction shows that $\dim ({\mathcal {Z}}({\mathcal {M}}))=\infty$.

(ii)$\Rightarrow$(i). Let $\tau '$ be an arbitrary faithful normal semifinite trace on ${\mathcal {M}}$. By the assumption, there exists a sequence of pairwise disjoint non-zero projections $\{E_n\}_{n=1}^\infty \subset {\mathcal {Z}}({\mathcal {M}})$ such that $\bigvee _{n=1}^\infty E_n=\mathbf{1 }$. In every algebra ${\mathcal {M}}_{E_n}$, there exists a non-zero finite projection $P_n$. If for some n we have that $\tau '(P_n)=\infty$, then the assertion follows from Remark 2.7.

Assume that $\tau '(P_n)<\infty$ for all n. Since $P_n\in {{\mathcal {M}}}_{E_n}$ and $E_n$ are pairwise disjoint, it follows that the central supports of $P_n$ are pairwise disjoint. Hence, $P:=\bigvee _{n=1}^\infty P_n$ is also a finite projections. Set $\tau (X):=\sum _{n=1}^\infty n \tau '(XE_n)/\tau '(P_n)$. Clearly, $\tau$ is a faithful normal semifinite trace on ${\mathcal {M}}$ and $\tau (P)=\infty$. By Remark 2.7, we obtain the validity of (i). $\quad \square$

3 Preliminaries on Derivations

In the present paper, we consider derivations $\delta$ from an arbitrary von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, that is, $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is a linear mapping satisfying $\delta (XY)=\delta (X)Y +X\delta (Y)$, $X,Y\in {{\mathcal {A}}}$. A derivation is called skew-adjoint if $\delta =-\delta ^*$, where $\delta ^*$ is a derivation defined by $\delta ^*(X) = (\delta (X^*))^*,~ x\in {{\mathcal {A}}}$. Actually, we can assume that the derivation $\delta$ is skew-adjoint because every derivation $\delta : {{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ can be decomposed into skew-adjoint components $\delta =\delta _1 +i\cdot \delta _2$, where

$$\begin{aligned} \delta _1 (X): =\frac{\delta (X) - \delta (X^*)^*}{2} \text{ and } \delta _2 (X):=\frac{\delta (X) +\delta (X^*)^*}{2i}. \end{aligned}$$

Remark 3.1

Assume that there exists an operator $T\in {{\mathcal {M}}}$ such that the skew-adjoint derivation $\delta =\delta _ T= [\cdot , T]$. For every $X\in {{\mathcal {A}}}$, we have

$$\begin{aligned} {[}X, T-T^*]=[X,T]-[X,T^*]=[X,T]+[X^*,T]^*=\delta (X)+(\delta (X^*))^*=0, \end{aligned}$$

which implies that ${{\,\mathrm{{\mathrm Im}}\,}}(T) = \frac{T-T^*}{2i}\in {{\mathcal {A}}}'$. Thus, for every $X\in {{\mathcal {A}}}$, we have

$$\begin{aligned} \delta (X)=[X,T]=[X,{{\,\mathrm{{\mathrm Re}}\,}}(T)+i{{\,\mathrm{{\mathrm Im}}\,}}(T)]=[X,{{\,\mathrm{{\mathrm Re}}\,}}(T)]. \end{aligned}$$

Hence, without loss of generality, we can always assume that the operator T implementing a skew-adjoint derivation $\delta$ is self-adjoint.

In the following, we need to consider several types of reductions of a given derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. The first one is reduction of $\delta$ by a given central projection in the algebra ${{\mathcal {M}}}$.

Lemma 3.2

Let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation and let $Z\in {{\mathcal {Z}}}({{\mathcal {M}}})$ be a projection. The mapping $\delta ^{(Z)} :{{\mathcal {A}}}_{Z} \rightarrow Z {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) Z ={{\mathcal {C}}}_0({{\mathcal {M}}}_{Z },\tau )$ given by $\delta ^{(Z)}(X Z ) = Z \delta (X)Z$, $X\in {{\mathcal {A}}}$, is a well-defined derivation from the induced von Neumann algebra ${{\mathcal {A}}}_Z$ into $Z {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) Z$.

Proof

If $A,B\in {{\mathcal {A}}}$ such that $AZ =BZ$, then

$$\begin{aligned}&\delta ^{(Z)}(AZ)- \delta ^{(Z)}(BZ)\\&\quad =Z\delta (A) Z - Z\delta (B) Z \\&\quad =Z \delta ((A-B)E^{|A-B|}(0,\infty ))Z \\&\quad =Z \delta (A-B)\cdot E^{|A-B|}(0,\infty ) Z +Z (A-B) \cdot \delta (E^{|A-B|}(0,\infty ) )Z\\&\quad =Z \delta (A-B) \cdot E^{|A-B|}(0,\infty )Z. \end{aligned}$$

Since $Z\in {{\mathcal {Z}}}({{\mathcal {M}}})$, it follows that $E^{|A-B|}(0,\infty )Z$ is a projection with $E^{|A-B|}(0,\infty )Z \le E^{|A-B|}(0,\infty )$. However, the assumption, $(A-B) E^{|A-B|}(0,\infty )Z =(A-B) Z =0$ implies that $E^{|A-B|}(0,\infty )Z =0$ and therefore $\delta ^{(Z)}(A Z) = \delta ^{(Z)}(B Z)$. Moreover, for every $X, Y\in {{\mathcal {A}}}$, we have

$$\begin{aligned} \delta ^{(Z)}(ZXZ \cdot ZYZ)&=\delta ^{(Z)}(ZXYZ) =Z\delta (XY)Z\\&= Z\delta (X)Y Z +ZX\delta (Y)Z\\&= Z\delta (X)Z\cdot ZY Z +ZXZ\cdot Z\delta (Y)Z\\&=\delta ^{(Z)}(ZXZ)ZY Z +ZXZ\delta ^{(Z)}(ZYZ), \end{aligned}$$

which implies that $\delta ^{(Z)}$ is a well-defined derivation. $\quad \square$

The other reduction of $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ we intend to use depends on the type of the algebra ${{\mathcal {A}}}$ with an additional assumption that $\delta |_{{{\mathcal {Z}}}({{\mathcal {A}}})}$ vanishes. For every $Z\in {{\mathcal {Z}}}({{\mathcal {A}}})$, the center of ${{\mathcal {A}}}$, the mapping $Z\delta (\cdot ) Z$ is a derivation from ${{\mathcal {A}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Moreover, if $\delta$ vanishes on ${{\mathcal {Z}}}({{\mathcal {A}}})$, then $Z\delta (\cdot ) Z$ is a derivation from ${{\mathcal {A}}}_{Z}$ into $Z {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z$, which coincides with $\delta (\cdot )$ on ${{\mathcal {A}}}_Z$. Let $Z_1,Z_2$ be two projections in ${{\mathcal {M}}}$ such that $Z_1Z_2=0$. For elements $X_1\in {{\mathcal {M}}}_{Z_1}=Z_1{{\mathcal {M}}}Z_1$ and $X_2\in {{\mathcal {M}}}_{Z_2}=Z_2{{\mathcal {M}}}Z_2$, we frequently denote $X_1+X_2$ by $X_1\oplus X_2$.

Lemma 3.3

Let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation such that $\delta |_{{{\mathcal {Z}}}({{\mathcal {A}}})}=0$. If for $Z_1 ,Z_2\in {{\mathcal {Z}}}({{\mathcal {A}}})$ with $Z_1Z_2=0$, $\delta |_{{{\mathcal {A}}}_{Z_1}}$ and $\delta |_{{{\mathcal {A}}}_{Z_2}}$ are inner derivations implemented by $T_1 \in Z_1{{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_1$ and $T_2 \in Z_2{{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_2$, then $\delta |_{{{\mathcal {A}}}_{Z_1+Z_2}}$ is implemented by $T_1 \oplus T_2$.

Proof

For every $X\in {{\mathcal {A}}}_{Z_1+Z_2}$, we have

$$\begin{aligned} \delta (X)&=\delta (XZ_1 +XZ_2) = \delta (XZ_1 )+\delta (XZ_2)= \delta _{T_1}(XZ_1 )+\delta _{T_2}(XZ_2)\\&= [XZ_1 ,T_1] + [XZ_2, T_2]= [X,T_1+T_2] =\delta _{T_1+T_2}(X), \end{aligned}$$

which completes the proof. $\quad \square$

Lemma 3.3 allows us to make the following reduction of the problem considered in this paper.

Remark 3.4

Let $P_1,P_2,P_3 \in {{\mathcal {Z}}}({{\mathcal {A}}})$ be the central partition of unity (some of $P_i$ can be zero), such that ${{\mathcal {A}}}_{P_1}$ is of type $I_{fin}$, ${{\mathcal {A}}}_{P_2}$ is of type II$_1$, ${{\mathcal {A}}}_{P_3}$ is properly infinite. Assume that $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ vanishes on ${{\mathcal {Z}}}({{\mathcal {A}}})$. By reducing $\delta$ to the algebras ${{\mathcal {A}}}_{P_i}$, $i=1,2,3$, to prove that $\delta$ is inner derivation, it is sufficient to consider separately the cases when ${{\mathcal {A}}}$ is type I, type II$_1$ or properly infinite. As we show in Sect. 4 (see Remark 4.4), the assumption that $\delta$ vanishes on ${{\mathcal {Z}}}({{\mathcal {A}}})$ can be imposed without loss of generality.

Next, we introduce a special subset $K_\delta$ of the algebra ${{\mathcal {M}}}$ generated by derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}$. As we prove later, for any derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, $K_\delta$ contains the operator implementing $\delta$.

Definition 3.5

For a skew-adjoint derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}$, we define by $K_\delta$ the weak$^*$ (or ultraweak) operator closure of $co\{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}$, where co(S) denotes the convex hull of a set S.

Remark 3.6

Recall that the strong operator closure, the weak operator closure and the weak$^*$ operator closure of the convex hull of a uniformly bounded set in ${{\mathcal {M}}}$ coincide (see e.g. [62, Chapter II, Lemma 2.5] and [20, Chapter IX, Corollary 5.2]). By Ringrose’s theorem [56] (see also [5, Theorem 3.1]), derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is bounded and therefore, the set $\{ \delta (U) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}$ is uniformly bounded. Thus,

$$\begin{aligned} K_\delta = {\overline{co}}^{wo} \{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}={\overline{co}}^{so} \{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}, \end{aligned}$$

(3.1)

where ${\overline{co}}^{so}(S)$ (respectively, ${\overline{co}}^{wo}(S)$) denotes the strong operator closure (respectively, weak operator closure) of convex hull of a set S. In particular, $\Vert X\Vert _\infty \le \Vert \delta \Vert _{{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}}$ for every $X\in K_\delta$. Furthermore, since $\delta$ is assumed to be skew-adjoint, using Leibniz rule, for any unitary $U\in {{\mathcal {A}}}$, we have

$$\begin{aligned} (U\delta (U^*))^* =-\delta (U)U^* = U\delta (U^* ) -\delta (\mathbf{1}) =U\delta (U^* ) , \end{aligned}$$

which implies that every element in $K_\delta$ is self-adjoint.

Remark 3.7

Let $Z_1,Z_2,\cdots , Z_n \in {{\mathcal {Z}}}({{\mathcal {A}}})$ be mutually disjoint projections such that $\delta (Z_i) =0$ for $i=1,2,\cdots ,n$. For every $Z_i$, we have

$$\begin{aligned} K_\delta Z_i&= {\overline{co}}^{wo}\{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} Z_i = {\overline{co}}^{wo}\{ U \delta (U^*) Z_i \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}\\&={\overline{co}}^{wo}\{ U Z_i\delta (U^*Z_i)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} ={\overline{co}}^{wo}\{ U \delta (U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}}_{Z_i})\}. \end{aligned}$$

Since $\delta (\mathbf{1}) =0$, it follows that $\delta (\mathbf{1} -\sum _{i=1}^n Z_i) =0$. Therefore, since $Z_i$ are mutually disjoint, for every $U_1,U_2,\cdots U_n \in {{\mathcal {U}}}({{\mathcal {A}}})$, we have that

$$\begin{aligned} \sum _{i=1}^n U_i Z_i \delta (U^*_i Z_i )&=\left( \sum _{i=1}^n U_i Z_i \right) \delta ( \sum _{i=1}^n U_i Z_i ) \\&= \left( \sum _{i=1}^n U_i Z_i +\mathbf{1} -\sum _{i=1}^n Z_i \right) \delta \left( \sum _{i=1}^n U_i Z_i + \mathbf{1} -\sum _{i=1}^n Z_i\right) . \end{aligned}$$

Note that $\sum _{i=1}^n U_i Z_i + \mathbf{1} -\sum _{i=1}^n Z_i\in {{\mathcal {U}}}({{\mathcal {A}}})$. Thus, $\sum _{i=1}^n U_i Z_i \delta (U^*_i Z_i ) \in K_\delta$. For any $X_1\in K_\delta Z_1$, there is a net in $co \{ U_1 Z _1 \delta (U^*_1 Z_1 ) \}$ converging to $X_1$ in the weak operator topology. Hence, $X_1 \oplus ( \oplus _{i=2}^n U_i Z_i \delta (U^*_i Z_i ) ) \in K_\delta$. By mathematical induction, we obtain that $\oplus _{i=1}^n X_i \in K_\delta$ for any $X_i\in K_\delta Z_i$. That is, $\sum _{i=1}^n K_\delta Z_i \subset K_\delta$.

In the following Proposition, we provide an auxiliary result which allows us to use Lemma 2.6 in the proof of Theorem 4.2.

Proposition 3.8

Let $T\in K_\delta$ and let $\varepsilon ,s >0$. If $0< s\le \tau (E^{|T|}(\varepsilon ,\infty ))$, then there is a unitary element $U\in {{\mathcal {U}}}({{\mathcal {A}}})$ such that

$$\begin{aligned} \int _0^{s/2} \mu ( t; \delta (U)) dt > \frac{s}{2} \varepsilon . \end{aligned}$$

Proof

Since $T\in K_\delta$, it follows from (3.1) that there is a net $\{B_\alpha \}_\alpha$ with

$$\begin{aligned} B_\alpha := \sum _{i=1}^{n_\alpha } \lambda _{\alpha }^{(i)} U_{ \alpha }^{(i)} \delta ((U_{ \alpha }^{(i)})^*) , 1\le n_\alpha <\infty ,~ U_{ \alpha }^{(i)} \in {{\mathcal {U}}}({{\mathcal {A}}}) , ~ \sum _{i=1}^{n_\alpha } \lambda _{\alpha }^{(i)} =1, \end{aligned}$$

converging to T in the strong operator topology. Note that every $B_\alpha$ is self-adjoint. By [52, Proposition 2.3.2], we have $|B_\alpha | \rightarrow _{so} |T|$, and therefore, employing Lemma 2.3, we infer that there exists a $B_\alpha$ such that $\tau (E^{|B_\alpha |}(\varepsilon ,\infty ))> \frac{s}{2}$. Hence, Lemma 2.4 implies that

$$\begin{aligned} \mu \Big (t; \sum _{i=1}^{n_\alpha } \lambda _{\alpha }^{(i)} U_{ \alpha }^{(i)} \delta ((U_{ \alpha }^{(i)})^*) \Big ) = \mu (t; B_\alpha )> \varepsilon , \quad t\in [0,\frac{s}{2} ]. \end{aligned}$$

(3.2)

Now, it follows from [47, Theorem 3.3.3] that

$$\begin{aligned} \sum _{i=1}^{n_\alpha } \lambda _{\alpha }^{(i)} \int _0^\frac{s}{2} \mu (t; U_{ \alpha }^{(i)} \delta ((U_{ \alpha }^{(i)})^*) )dt\ge \int _0^\frac{s}{2} \mu (t; \sum _{i=1}^{n_\alpha } \lambda _{\alpha }^{(i)} U_{ \alpha }^{(i)} \delta ((U_{ \alpha }^{(i)})^*) )dt {\mathop {>}\limits ^{(3.2)}} \frac{s}{2}\varepsilon . \end{aligned}$$

Thus, there exists $U_{\alpha }^{(i)} \in {{\mathcal {U}}}({{\mathcal {A}}})$ such that

$$\begin{aligned} \int _0^\frac{s}{2} \mu (t; \delta ((U_{\alpha }^{(i)})^*) ) dt {\mathop {\ge }\limits ^{(2.2)}} \int _0^\frac{s}{2} \mu (t; U_{\alpha }^{(i)} \delta ((U_{\alpha }^{(i)})^*) ) dt > \frac{s}{2} \varepsilon . \end{aligned}$$

$\square$

Recall, that our aim is to show that any derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ inner. Hence, if we have a central partition of unity $\{Z_i\}$ of ${{\mathcal {A}}}$ such that $\delta$ is inner on every ${{\mathcal {A}}}_{Z_i}$ and is implemented by $K_i\in Z_i{{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_i$, then a natural choice of element implementing $\delta$ on ${{\mathcal {A}}}$ is $\oplus _i K_i$. However, it can happen that $K_i \in Z_i {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) Z_i$, but $\oplus _i K_i \notin {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ (as an example, consider the algebra ${{\mathcal {M}}}=L_\infty (0,\infty )$ and partition $\{Z_i\} = \{\chi _{(i,i+1]}\}$). The latter fact is in direct contrast with [55, 2.11.], since if $\{Z_i\}_i$ is a central partition of the identity of ${{\mathcal {M}}}$, then the direct sum of a family of uniformly bounded operators $K_i \in {{\mathcal {J}}}({{\mathcal {M}}}_{Z_i})$ is also in ${{\mathcal {J}}}({{\mathcal {M}}})$. We tackle this issue for $\tau$-compact operators, by showing that under additional assumption that every operator $K_i$ is chosen from $Z_iK_\delta$, the direct sum $K:=\oplus _i K_i$ is also $\tau$-compact.

Theorem 3.9

Let ${{\mathcal {A}}}$ be a von Neumann subalgebra of ${{\mathcal {M}}}$ and let $\{Z_i\in {{\mathcal {P}}}({{\mathcal {A}}})\}_i$ be a central partition of the unity in ${{\mathcal {A}}}$. Assume that $\delta (Z)=0$ for every $Z\in {{\mathcal {Z}}}({{\mathcal {A}}})$. If there exists $K_i\in {{\mathcal {C}}}_0({{\mathcal {M}}}_{Z_i},\tau ) \cap Z_i K_\delta$ such that $\delta =\delta _{K_i}$ on ${{\mathcal {A}}}_{Z_i}$ for every i, then $K:=\oplus _i K_i\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\cap K_\delta$ with $\delta =\delta _K$ on ${{\mathcal {A}}}$.

Proof

Note that the operators $K_i$ and K are self-adjoint. Since $\delta (Z_i) =0$ for every i, it follows that $Z_i\delta (X) =\delta (Z_iXZ_i) =Z_i\delta (X)Z_i=\delta (X)Z_i$ for every $X\in {{\mathcal {A}}}$. Hence, that fact that $\{Z_i\}$ is a central partition of unity, together with the assumption that $\delta =\delta _{K_i}$ on ${{{\mathcal {A}}}_{Z_i}}$ implies that for every $X\in {{\mathcal {A}}}$, we have

$$\begin{aligned} \delta (X)&= \oplus _i (Z_i \delta (X)) = \oplus _i (Z_i \delta (X) Z_i)= \oplus _i \delta (Z_iXZ_i )\\&=\oplus _i \delta _{K_i}(Z_i XZ_i) =\oplus _i \delta _{K_i}( X) = \delta _{K}( X). \end{aligned}$$

We assert that $K\in K_\delta$. Let $\tau _{{{\mathcal {Z}}}({{\mathcal {A}}})}$ be a semifinite faithful normal trace ${{\mathcal {Z}}}({{\mathcal {A}}})$. It follows that there is an increasing net $\{R_\lambda \} \subset \{{{\mathcal {Z}}}({{\mathcal {A}}})\}$ of $\tau _{{{\mathcal {Z}}}({{\mathcal {A}}})}$-finite projections such that $R_\lambda \uparrow \mathbf{1}$. It is a fact that the reduced von Neumann algebra ${{\mathcal {Z}}}({{\mathcal {A}}})_{R_\lambda }$ is finite and countably decomposable for every $R_\lambda$ (see e.g. [51, Theorem 1.3.6] for a proof of this fact). Thus, for every fixed $\lambda$, there are only countably many $Z_i$ such that $Z_iR_\lambda \ne 0$. We denote the sequence consists of non-zero elements from $\{Z_i R_\lambda \}$ by $\{P_n \}_{n=1}^\infty$. Note that for every k, we have

$$\begin{aligned} P_k K P_k \in P_k K_\delta P_k . \end{aligned}$$

By Remark 3.7, $\oplus _{k=1}^n P_k K P_k\in K_\delta R_\lambda$ for every n. Since $\sum _{k=1}^\infty P_k = R_\lambda$, it follows that

$$\begin{aligned} R_\lambda K R_\lambda = \oplus _{k=1}^\infty P_kK P_k \in K_\delta R_\lambda . \end{aligned}$$

Since $R_\lambda \in {{\mathcal {Z}}}({{\mathcal {A}}})$, by Remark 3.7 again, we have that

$$\begin{aligned} R_\lambda K R_\lambda \in K_\delta R_\lambda \subset K_\delta . \end{aligned}$$

Since $R_\lambda KR_\lambda \rightarrow _{so} K$, we obtain that $K\in K_\delta$.

Now, we prove that K is $\tau$-compact. If the net $\{Z_i\}$ consists of finitely many projections, then K is clearly $\tau$-compact. We assume that $\{Z_i\}$ contains infinitely many projections and $K\notin {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. By the definition of ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, there exists an $\varepsilon >0$ such that $\infty = \tau (E^{|K|}(\varepsilon ,\infty )) = \tau (E^{\oplus _i |K_i |}(\varepsilon ,\infty ))$. Noting that $\tau$ is completely additive (see e.g. [63, Chapter VII, Theorem 1.11]), we obtain that $\sum _i \tau (E^{|K_i |}(\varepsilon ,\infty ))=\infty$. Hence, we can choose countably many distinct $T_j:= K_{i(j)}$ from $\{K_i\}$ such that

$$\begin{aligned} \tau (E^{\oplus _{j=1}^\infty |T_j|}(\varepsilon ,\infty ))=\sum _{j=1}^\infty \tau (E^{ |T_j |}(\varepsilon ,\infty ))=\sum _{j=1}^\infty t_j =\infty , \end{aligned}$$

where $t_j:= \tau (E^{|T_j|}(\varepsilon ,\infty ))\in (0,\infty )$, $1 \le j<\infty$. We denote $Z_{i(j)}$ by $Q_j$.

Note that for every $1\le j < \infty$, we have

$$\begin{aligned} T_j&\in {\overline{co}}^{so} \{ U \delta (U^*) | U\in {{\mathcal {U}}}({{\mathcal {A}}}_{Q_j}) \}. \end{aligned}$$

For every j, by Proposition 3.8, we can choose a $U_j \in {{\mathcal {U}}}( {{\mathcal {A}}}_{Q_j})$ such that

$$\begin{aligned} \int _0^{\frac{t_j}{2}} \mu (t; \delta (U_j) ) dt > \frac{t_j }{2}\varepsilon . \end{aligned}$$

(3.3)

Let $U:= \oplus _{j=1}^\infty U_j \in {{\mathcal {A}}}$. Since $\delta$ vanishes on $\{Q_j\}$, it follows that $\delta (U_j)=\delta (Q_jU Q_j) =Q_j\delta (U)Q_j$. Thus, for every n, we have

$$\begin{aligned} \int _0^{\sum _{j=1}^n \frac{t_j }{2}} \mu (t;\delta (U)) dt {\mathop {\ge }\limits ^{(2.3)}} \sum _{j=1}^n \int _0^{ \frac{t_j }{2}} \mu (t;\delta (U_j) ) dt{\mathop {\ge }\limits ^{(3.3)}} \sum _{j=1}^n \frac{t_j }{2}\varepsilon . \end{aligned}$$

Noticing that $\delta (U)\in {{\mathcal {M}}}$ and recalling that $\sum _{j=1}^\infty t_j=\infty$, by Lemma 2.6, we obtain that $\delta (U)$ is not $\tau$-compact, which is a contradiction and hence $K \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. $\square$

We end this section by showing a fine property of inner derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, which is related to the set $K_\delta$ (see Definition 3.5). As we show in Proposition 3.10 below, for any inner derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, there exists an operator $T'\in K_\delta$ implementing $\delta$. We note that analogous property for inner derivations from ${{\mathcal {A}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ is established in [55], however, our approach is completely different from that used in the proof of [55, Lemma 4.6]. Furthermore, our result holds if the assumption on ${{\mathcal {A}}}$ is relaxed to a weaker assumption that ${{\mathcal {A}}}$ is a unital (that is, $\mathbf{1}_{{\mathcal {A}}}=\mathbf{1}_{{\mathcal {M}}}$) $C^*$-subalgebra of ${{\mathcal {M}}}$.

Proposition 3.10

Let ${{\mathcal {N}}}$ be a unital $C^*$-subalgebra of ${{\mathcal {M}}}$ and let $\delta :{{\mathcal {N}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation. If there exists $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ such that $\delta =\delta _T$, then there exists an element $T'\in K_\delta ={\overline{co}}^{wo} \{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {N}}}) \}$ such that $\delta =\delta _{T'}$.

Proof

Let $P_n :=E^{|T|}(\frac{1}{n},\infty )$ and let $T_n := TP_n$. For every n, projection $P_n$ is $\tau$-finite and $\Vert T-T_n\Vert _\infty \le \frac{1}{n}$. In particular, $T_n \in L_2({{\mathcal {M}}},\tau )$, where $L_2({{\mathcal {M}}},\tau )$ denotes the noncommutative $L_2$-space affiliated with ${{\mathcal {M}}}$. Hence, $\delta _{T_n}$ has range inside $L_2({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$ and therefore, by [10, Theorem 3.1 and Proposition 3.4], there exists

$$\begin{aligned} T_n' \in \overline{co\{ U \delta _{T_n}(U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {N}}})\}}^{\Vert \cdot \Vert _2} =\overline{co\{ T_n - U T_n U^* \mid U\in {{\mathcal {U}}}({{\mathcal {N}}})\}}^{\Vert \cdot \Vert _2} \end{aligned}$$

such that $\Vert T_n'\Vert _\infty \le \Vert \delta _{T_n}\Vert _\infty \le 2 \Vert T_n\Vert _\infty \le 2\Vert T\Vert _\infty$ and $\delta _{T_n} =\delta _{T_n'}$. Hence, by [20, Chapter IX, Proposition 5.5], there is a (wo)-cluster point $T'\in \cap _n\overline{\{T_n', T_{n+1}' ,\cdots \}}^{wo}$ for the sequence $\{T_n'\}$ in the ball of radius $2\Vert T\Vert _\infty$ in ${{\mathcal {M}}}$.

Since $\Vert \cdot \Vert _{2}$ induces the strong operator topology, and the strong operator closure and the weak operator closure of the convex hull of a uniformly bounded set coincide, it follows that $T'_n \in {\overline{co}}^{wo}\{ T_n - U T_n U^* \mid U\in {{\mathcal {U}}}({{\mathcal {N}}})\}$. Since $\Vert T_n -UT_n U^* -(T- U T U^*)\Vert _\infty \le \frac{2}{n}$, it follows from the Kaplansky density theorem (see e.g. [62, Chapter II, Theorem 4.8]) that there is an element $B_n\in {\overline{co}}^{wo}\{ T - U T U^* \mid U\in {{\mathcal {U}}}({{\mathcal {N}}})\}=K_\delta$ such that

$$\begin{aligned} \Vert T_n' -B_n \Vert _\infty \le \frac{2}{n} .\end{aligned}$$

Thus, $T'$ is a (wo)-cluster point of $\{B_n\}$ and therefore $T'\in K_\delta$.

For every $X\in {{\mathcal {N}}}$, $\eta , \xi \in {{\mathcal {H}}}$, we set $\omega (\cdot ) =\langle \cdot X\eta , \xi \rangle$ and $\rho (\cdot ) = \langle \cdot \eta , X^* \xi \rangle$ on ${{\mathcal {M}}}$. For every $\varepsilon >0$, there exists $N >2/\varepsilon$ such that

$$\begin{aligned} |\omega (T' -T'_N)|, \ |\rho (T'-T'_N)| <\varepsilon . \end{aligned}$$

Recall that $\delta _{T_N}=\delta _{T_N'}$. We have

$$\begin{aligned}&| \langle [T-T', X]\eta ,\xi \rangle | \\&\quad \le | \langle [T-T_N, X]\eta ,\xi \rangle | +|\langle [T_N- T_N', X]\eta ,\xi \rangle |+|\langle [T_N'- T', X] \eta ,\xi \rangle |\\&\quad \le | \langle [T-T_N, X]\eta ,\xi \rangle | + |\langle [T_N'- T', X] \eta ,\xi \rangle | \\&\quad \le |\langle (T-T_N) X\eta , \xi )\rangle |+ |\langle X (T-T_N) \eta , \xi )\rangle | \\&\quad \quad + |\omega (T_N'- T')| + |\rho (T_N'- T')| \\&\quad \le \frac{2}{N}\Vert X\Vert _\infty \Vert \eta \Vert _{{\mathcal {H}}}\Vert \xi \Vert _{{\mathcal {H}}}+ 2\varepsilon \le \varepsilon \Vert X\Vert _\infty \Vert \eta \Vert _{{\mathcal {H}}}\Vert \xi \Vert _{{\mathcal {H}}}+ 2\varepsilon . \end{aligned}$$

Since $\varepsilon$ is arbitrary, we infer that $[T-T', X]=0$ for every $X\in {{\mathcal {N}}}$. Hence, $T-T' \in {{\mathcal {N}}}'$ and therefore $\delta =\delta _T =\delta _{T'}$. $\square$

4 The Abelian Case

Let ${{\mathcal {M}}}$ be a von Neumann algebra equipped with a semi-finite faithful normal trace $\tau$ and let ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be the ideal of all $\tau$-compact operator affiliated with ${{\mathcal {M}}}$. In this section, we consider derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, where ${{\mathcal {A}}}$ is an abelian von Neumann subalgebra of ${{\mathcal {M}}}$. We show that in this case, any derivation $\delta$ is inner. In particular, this result allows us to assume in the following sections that we work with derivations vanishing on the center of the subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$.

We note that even though ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {J}}}({{\mathcal {M}}})$ and ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ behaves somewhat like ${{\mathcal {J}}}({{\mathcal {M}}})$, the additional restrictions to the abelian subalgebra ${{\mathcal {A}}}$ in [44, 55] are no longer required. Moreover, since ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is not necessarily the dual space of a Banach space, the techniques used in [44, Theorem 14] are not applicable in this case.

Throughout this section, we assume that ${{\mathcal {A}}}$ is an abelian von Neumann subalgebra of ${{\mathcal {M}}}$.

Let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}$ be a derivation. The following result is well-known (see e.g. [44, Section 3] and [37, Theorem 2.1]).

Proposition 4.1

If ${{\mathcal {A}}}$ is an abelian von Neumann subalgebra of ${{\mathcal {M}}}$, then every derivation $\delta$ from ${{\mathcal {A}}}$ into ${{\mathcal {M}}}$ is inner. That is, $\delta =\delta _T$ for some $T\in {{\mathcal {M}}}$.

In what follows, we consider derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Since ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {M}}}$, $\delta$ is a derivation from ${{\mathcal {A}}}$ into ${{\mathcal {M}}}$ and therefore there exists an operator $T\in {{\mathcal {M}}}$ such that $\delta =\delta _ T$. Thus, our aim in this section is to show that T can be chosen to be $\tau$-compact.

Recall that an expectation $\Phi$ is a norm one projection from $B({{\mathcal {H}}})$ onto a von Neumann algebra (see [21, Section 8]). Motivated by the idea related to an expectation from $B({{\mathcal {H}}})$ onto ${{\mathcal {A}}}'$ used in [21, Theorem 10.9], we prove the main theorem of this section by techniques different from those used in [44], extending the results in [44] to the case of ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$.

Theorem 4.2

Assume that ${{\mathcal {A}}}$ is an abelian von Neumann subalgebra of ${{\mathcal {M}}}$. Every derivation $\delta : {{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner, that is, $\delta =\delta _K$ for some $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\cap K_\delta$.

Proof

Without loss of generality, we may assume that $\delta$ is a skew-adjoint derivation from ${{\mathcal {A}}}$ into ${{\mathcal {M}}}$ (see Sect. 3). Proposition 4.1 guarantees that there exists $T\in {{\mathcal {M}}}$ such that $\delta =\delta _T$. In particular, we may assume that T is self-adjoint (see Remark 3.1). Let $\Phi$ be an expectation from $B({{\mathcal {H}}})$ onto ${{\mathcal {A}}}'$ given by [21, Theorem 8.3]. By the construction of $\Phi$ (see [21, Theorem 8.3]), we have that $\Phi (T)$ belongs to the weak$^*$ operator closed convex hull of $\{UTU^*: U\in {{\mathcal {U}}}({{\mathcal {A}}})\}$. In particular, $\Phi (T)\in {{\mathcal {M}}}$. Set $K := T-\Phi (T) \in {{\mathcal {M}}}$. It is clear that $\delta =\delta _K$ and K belongs to the weak$^*$ operator closure of the convex hull of $\{U\delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\}$. It suffices to prove that $K \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Since T is self-adjoint, it follows that K is also self-adjoint.

Assume by contradiction that $K\notin {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, i.e., there is an $\varepsilon >0$ such that $\mu (\infty ;K)> \varepsilon$. We claim that there exists $A\in {{\mathcal {A}}}$ such that $\delta (A)$ is not $\tau$-compact. To this end, we intend to use Lemma 2.6. For convenience, we divide the proof into several steps.

(a) Let

$$\begin{aligned} {{\mathcal {P}}}:= \{ P\in {{\mathcal {P}}}({{\mathcal {A}}}) \mid \mu (\infty ;PKP) > \varepsilon \}. \end{aligned}$$

We claim that there is a maximal downwards directed chain $\{P_\gamma \}$ of infinitely many elements in ${{\mathcal {P}}}$ which satisfies $P_0:=\inf \{P_\gamma \}\notin {{\mathcal {P}}}$ and $P_\gamma -P_0 \in {{\mathcal {P}}}$ for every $\gamma .$

It is clear that ${{\mathcal {P}}}$ is not empty as $\mathbf{1 }\in {{\mathcal {P}}}$. We note, in addition, that $\tau (P)=\infty$ for any $P\in {{\mathcal {P}}}$. Take an arbitrary $P\in {{\mathcal {P}}}$. Assume that P is minimal in ${{\mathcal {A}}}$. The same argument in the proof of [44, Lemma 8 or Theorem 14] yields that $PTP =0$. Since $P\in {{\mathcal {A}}}\subset {{\mathcal {A}}}'$, it follows from [21, Theorem 8.1] that

$$\begin{aligned} PKP = PTP -P\Phi (T)P =PTP -\Phi (PTP)=0, \end{aligned}$$

which is a contradiction to $P\in {{\mathcal {P}}}$. Thus, ${{\mathcal {P}}}$ contains no minimal element in ${{\mathcal {A}}}$.

Now, let $Q\in {{\mathcal {P}}}({{\mathcal {A}}})$ be such that $0\ne Q \lneqq P$ and let $Q_1= Q, Q_2=P-Q$. We have

$$\begin{aligned} PKP&=Q_1KQ_1 +Q_2KQ_2 + Q_1K Q_2 +Q_2KQ_1 \nonumber \\&=Q_1KQ_1 +Q_2KQ_2 + \delta (Q_1)Q_2 + \delta (Q_2)Q_1, \end{aligned}$$

(4.1)

where we used the fact that $Q_1\perp Q_2$ and $\delta =\delta _K$ for the second equality. Since $\mu (\infty ;PKP)> \varepsilon$, [47, Corollary 2.3.16] implies that

$$\begin{aligned}&\mu (t;Q_1KQ_1 +Q_2KQ_2 ) + \mu (s_1; \delta (Q_1)Q_2) +\mu (s_2; \delta (Q_2)Q_1) \\&\quad \ge \mu (t+s_1+s_2;PKP) \ge \mu (\infty ;PKP) > \varepsilon \end{aligned}$$

for all $t,s_1,s_2>0$. Let $\varepsilon _1$ be such that $\mu (\infty ;PKP)> \varepsilon _1> \varepsilon$. Since $\mu (s_1; \delta (Q_1)Q_2), \mu (s_2; \delta (Q_2)Q_1) \rightarrow 0$ as $s_1,s_2\rightarrow \infty$, we have

$$\begin{aligned} \mu (t; Q_1K Q_1 +Q_2K Q_2)> \varepsilon _1> \varepsilon ,~ t\in [0,\infty ). \end{aligned}$$

(4.2)

Assume that both projections $E^{|Q_1KQ_1|}(\varepsilon _1,\infty )$ and $E^{|Q_2KQ_2|}(\varepsilon _1,\infty )$ are $\tau$-finite. Since $Q_1 \perp Q_2$, it follows that $E^{|Q_1KQ_1|}(\varepsilon _1,\infty )+E^{|Q_2KQ_2|}(\varepsilon _1,\infty )\!=\!E^{|Q_1KQ_1+Q_2KQ_2|} (\varepsilon _1,\infty )$, and therefore

$$\begin{aligned} \tau (E^{|Q_1KQ_1+Q_2KQ_2|}(\varepsilon _1,\infty ))&= \tau (E^{|Q_1KQ_1| + |Q_2KQ_2|}(\varepsilon _1,\infty )) \\&= \tau (E^{|Q_1KQ_1|}(\varepsilon _1,\infty )) + \tau (E^{|Q_2KQ_2|}(\varepsilon _1,\infty )) <\infty . \end{aligned}$$

By Lemma 2.4, we obtain a contradiction to (4.2). Hence, either $\tau (E^{|Q_1KQ_1|}(\varepsilon _1,\infty ))=\infty$ or $\tau (E^{|Q_2KQ_2|}(\varepsilon _1,\infty )) =\infty$. By Lemma 2.4, either $\mu (\infty ; Q_1KQ_1)\ge \varepsilon _1>\varepsilon$ or $\mu (\infty ; Q_2KQ_2)\ge \varepsilon _1>\varepsilon$, which implies that either $Q_1$ or $Q_2$ belongs to ${{\mathcal {P}}}$. This shows that ${{\mathcal {P}}}$ has no minimal elements, that is, for every element $P\in {{\mathcal {P}}}$, we can always find an element $Q\in {{\mathcal {P}}}$ such that $Q\le P$. Moreover, if $P\in {{\mathcal {P}}}$ and $Q\in {{\mathcal {P}}}({{\mathcal {A}}})$ such that $Q\le P$, then either Q or $P-Q$ belongs to ${{\mathcal {P}}}$.

Let $\{P_\gamma \}$ be a maximal downwards directed chain in ${{\mathcal {P}}}$ and let $P_0 = \inf \{P_\gamma \}$. Obviously, $P_0\notin {{\mathcal {P}}}$. Otherwise, there exists a $P\lneqq P_0$ with $P\in {{\mathcal {P}}}$, which contradicts the maximality of $\{P_\gamma \}$. By the property stated in the above paragraph, either $P_\gamma -P_0$ or $P_0$ must belong to ${{\mathcal {P}}}$. However, $P_0\notin {{\mathcal {P}}}$. Thus, $P_\gamma -P_0 \in {{\mathcal {P}}}$.

(b) Now, let us construct a sequence $\gamma _1 \succ \gamma _2 \succ \cdots$ such that the projection $Q_k := P_{\gamma _k} - P_{\gamma _{k+1}}$ satisfies

$$\begin{aligned} \mu (t; Q_k K Q_k) >\varepsilon , ~ t\in [0,2]. \end{aligned}$$

(4.3)

Take an arbitrary $\gamma$ and set $\gamma _1=\gamma$. Assume that the sequence $\gamma _1 \prec \gamma _2 \prec \cdots \prec \gamma _n$ is constructed for some $n\in {\mathbb {N}}$. Let $A_{n}: = (P_{\gamma _{n}}- P_0) K (P_{\gamma _{n}}- P_0)$. We have that $A_{n}^*=A_{n}$. Furthermore, since $P_\gamma -P_0\in {{\mathcal {P}}}$, it follows that $\mu (\infty ; A_n)> \varepsilon$, which guarantees that $\tau (E^{|A_n|}(\varepsilon ,\infty )) = \infty$ (see Lemma 2.4). Since $\mathbf{1 } - P_\gamma +P_0 \uparrow \mathbf{1 }$, it follows from [52, Proposition 2.3.2] that

$$\begin{aligned} so-\lim _\gamma | (\mathbf{1 } -P_{\gamma } +P_0) A_{n}(\mathbf{1 } -P_{\gamma } +P_0) |= | A_{n}|. \end{aligned}$$

Then, by Lemma 2.3, we have

$$\begin{aligned} \liminf _{\gamma } \tau (E^{| (\mathbf{1 } -P_{\gamma } +P_0) A_{n}(\mathbf{1 } -P_{\gamma } +P_0) |}(\varepsilon , \infty )) \ge \tau (E^{|A_n|}(\varepsilon ,\infty ))=\infty . \end{aligned}$$

Hence, we can find $\gamma _{n+1}\succ \gamma _n$ such that $\tau ( E^{ | (\mathbf{1 } -P_{\gamma _{n+1}} +P_0) A_{n}(\mathbf{1 } -P_{\gamma _{n+1}} +P_0) |} (\varepsilon ,\infty ) ) >2$, and therefore, by Lemma 2.4, we have

$$\begin{aligned} \mu (t; (\mathbf{1 } -P_{\gamma _{n+1}} +P_0) A_{n}(\mathbf{1 } -P_{\gamma _{n+1}} +P_0) )> \varepsilon , ~t\in [0,2]. \end{aligned}$$

(4.4)

Since $P_\gamma \downarrow$, it follows that $(P_{\gamma _n} -P_0)(\mathbf{1} -P_{\gamma _{n+1}} +P_0 ) = (P_{\gamma _n } -P_{\gamma _{n+1}})$, which implies that, setting $Q_{n} := P_{\gamma _{n}} -P_{\gamma _{n+1}}$, we obtain that $\mu ( t; Q_{n} KQ_{n} )> \varepsilon$ for all $t\in [0,2 ]$.

(c) We claim that for every $k\in {\mathbb {N}}$, there is a $U_k \in {{\mathcal {U}}}({{\mathcal {A}}})$ such that

$$\begin{aligned} \int _0^1 \mu (t; Q_k \delta (U_{k}) Q_k ) dt > \varepsilon . \end{aligned}$$

(4.5)

Since $\Phi (K) = \Phi (T-\Phi (T)) = 0$, by [21, Theorem 8.3], the operator

$$\begin{aligned} Q_k K Q_k = Q_k (K-\Phi (K)) Q_k \end{aligned}$$

belongs to the weak$^*$ operator closure of

$$\begin{aligned} co\{ Q_k (K-UKU^*)Q_k:U\in {{\mathcal {U}}}({{\mathcal {A}}}) \} = co\{ Q_{k} U \delta (U^*) Q_k : U\in {{\mathcal {U}}}({{\mathcal {A}}})\}. \end{aligned}$$

Since ${{\mathcal {A}}}$ is abelian, it follows that $Q_k\delta (\cdot )Q_k$ is a derivation from ${{\mathcal {A}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ (see Sect. 3). Thus, $Q_k K Q_k \in K_{Q_k \delta (\cdot )Q_k }$. By the construction of $Q_k$ and Proposition 3.8, we conclude that $U_k\in {{\mathcal {U}}}({{\mathcal {A}}})$ satisfying (4.5) exists.

(d) Finally, since $Q_k,U_k\in {{\mathcal {A}}}$, $Q_i\perp Q_j$ ($i\ne j$) and ${{\mathcal {A}}}$ is abelian, the series $\sum _{k=1}^\infty Q_k U_k$ converges in ${{\mathcal {A}}}$ in the strong operator topology. We define

$$\begin{aligned} A:= \sum _{k=1}^\infty Q_k U_k. \end{aligned}$$

Since ${{\mathcal {A}}}$ is abelian, it follows that

$$\begin{aligned} Q_k\delta (Q_k^\perp X)Q_k =Q_k \delta (Q_k^\perp X Q_k^\perp )Q_k=Q_k Q_k^\perp X\delta ( Q_k^\perp )Q_k+Q_k \delta (Q_k^\perp X)Q_k^\perp Q_k =0 \end{aligned}$$

for every $X\in {{\mathcal {A}}}$, and therefore,

$$\begin{aligned} Q_k\delta (A)Q_k&=Q_k\delta (Q_k A)Q_k +Q_k \delta (Q_k ^\perp A)Q_k \nonumber \\&= Q_k \delta (Q_k U_k )Q_k \nonumber \\&= Q_k \delta (Q_k U_k) Q_k + Q_k \delta (Q_k^\perp U_k) Q_k\nonumber \\&= Q_k \delta (U_k) Q_k. \end{aligned}$$

(4.6)

Hence, we obtain

$$\begin{aligned} \int _0^1 \mu (t;Q_k\delta (A)Q_k) dt {\mathop {=}\limits ^{(4.6)}} \int _0^1 \mu (t;Q_k \delta (U_k) Q_k ) dt {\mathop {>}\limits ^{(4.5)}} \varepsilon . \end{aligned}$$

(4.7)

Take an arbitrary $n\ge 1$. Since $Q_i\perp Q_j$ for $i\ne j$, it follows that

$$\begin{aligned} \int _0^n \mu (t;\delta (A))dt {\mathop {\ge }\limits ^{(2.3)}} \sum _{i=1}^n \int _0^1 \mu (t; Q_i\delta (A) Q_i )dt {\mathop {>}\limits ^{(4.7)}} n\cdot \varepsilon . \end{aligned}$$

Now, by Lemma 2.6, we obtain that $\delta (A)$ is not $\tau$-compact, which is a contradiction. It completes the proof. $\square$

Remark 4.3

Note that the so-called locally compatible condition on the abelian von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ is required in studying derivations from ${{\mathcal {A}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ (see [55, Proposition 4.3], see also [44]). When this condition is not fulfilled, derivations from ${{\mathcal {A}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ are not necessarily inner (see [55, Theorem 1.2]). However, the “locally compatible” condition is redundant in our present setting, that is, the result of Theorem 4.2 holds without any additional assumption on the abelian subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$.

Remark 4.4

Assume that ${{\mathcal {A}}}$ is a von Neumann subalgebra of ${{\mathcal {M}}}$. By Theorem 4.2, for every derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, $\delta |_{{{\mathcal {Z}}}({{\mathcal {A}}})}$ is implemented by a $\tau$-compact operator K. Hence, in the study of derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, we can consider linear mapping $\delta -\delta _K$ which is a derivation from ${{\mathcal {A}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ vanishing on ${{\mathcal {Z}}}({{\mathcal {A}}})$. That is, without loss of generality, we may assume that derivation $\delta$ vanishes on ${{\mathcal {Z}}}({{\mathcal {A}}})$.

5 The Properly Infinite Case

Let, as before, ${{\mathcal {M}}}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$, and let ${{\mathcal {A}}}$ be a properly infinite von Neumann subalgebra of ${{\mathcal {M}}}$. Derivations acting on a properly infinite von Neumann algebra have been actively investigated. Christensen [15, Corollary 5.6] showed that every derivation from ${{\mathcal {A}}}$ into $B({{\mathcal {H}}})$ is inner. It is proved by Christensen, Effros and Sinclair [17, Corollary 5.5] that every derivation $\delta$ from ${{\mathcal {A}}}$ into ${{\mathcal {M}}}$ is inner if ${{\mathcal {M}}}$ is injective (see also [15, Corollary 5.6]). Recently, in [9], Ber, Chilin and Sukochev proved that every derivation $\delta$ from ${{\mathcal {A}}}$ into a Banach ${{\mathcal {A}}}$-bimodule of locally measurable operators affiliated with ${{\mathcal {A}}}$ is inner (see also [5, Theorem 4.7]).

In this section, we show that any derivation $\delta$ on a properly infinite von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ with values in the ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is necessarily inner. We note, that the result for derivations from an abelian subalgebra of ${{\mathcal {M}}}$ (see Theorem 4.2) allows us to use the same approach for properly infinite algebras as in [44] (see also [37] and [21]).

Recall that if ${{\mathcal {A}}}$ is properly infinite von Neumann subalgebra of a semifinite von Neumann algebra ${{\mathcal {M}}}$, then there is an infinite countable decomposition of the identity into mutually orthogonal projections of ${{\mathcal {A}}}$, all equivalent in ${{\mathcal {A}}}$ to $\mathbf{1 }$, and thus a fortiori equivalent in ${{\mathcal {M}}}$ to $\mathbf{1 }$ [22, Part III, Chapter 8, Section 6, Corollary 2] (see also [42]).

Let $H_0 = \ell ^2({\mathbb {Z}})$. By [22, Part I, Section 2.4, Proposition 5], there is a spatial isomorphism

$$\begin{aligned} \phi :{{\mathcal {M}}}\rightarrow {\tilde{{{\mathcal {M}}}}} = {{\mathcal {M}}}\otimes B(H_0)\end{aligned}$$

(5.1)

with

$$\begin{aligned} \phi ({{\mathcal {A}}}) = {\tilde{{{\mathcal {A}}}}} = {{\mathcal {A}}}\otimes B(H_0). \end{aligned}$$

It is well-known [22, Section 1.5, Proposition 8 ] that a spatial isomorphism is isometric and is normal, i.e., for every bounded increasing net $\{X_i \in {{\mathcal {M}}}_+ \}_i$ satisfying $X_i \uparrow X$, we have $\phi (X_i) \uparrow \phi (X)$. Recall also that the elements $B\in {\tilde{{{\mathcal {M}}}}}$ (or ${\tilde{{{\mathcal {A}}}}}$) are represented by matrices $[B_{ij}]$, $i,j\in {\mathbb {Z}}$, with entries in ${{\mathcal {M}}}$(or ${{\mathcal {A}}}$) by the formula

$$\begin{aligned} (\mathbf{1 }\otimes E_{ij})B(\mathbf{1 }\otimes E_{kl}) = B_{jk} \otimes E_{il}, \end{aligned}$$

where $E_{ij}$ is the canonical matrix unit of $B(H_0)$. In particular, if ${{\mathcal {L}}}$ (respectively, ${{\mathcal {D}}}$) is the maximal abelian subalgebras of $B(H_0)$ of Laurent (respectively, diagonal) matrices, then $B\in {{\mathcal {M}}}\otimes {{\mathcal {L}}}$ (respectively, $B\in {{\mathcal {M}}}\otimes {{\mathcal {D}}}$) if and only if $[B_{ij}]$ is a Laurent (respectively, a diagonal) matrix with entries in ${{\mathcal {M}}}$, i.e., $B_{ij} = B_{i-j}$ (respectively, $B_{ij} = \delta _{ij}B_{ii}$, where $\delta _{ij}$ stands for the Kronecker Delta), $i,j \in {\mathbb {Z}}$, where $B_k$ denotes the entry along the kth diagonal for all $k \in {\mathbb {Z}}$.

Let $\tau _0$ be the standard trace on $B(H_0)$ and ${\tilde{\tau }}: = \tau \otimes \tau _0$. For the properties of tensor products of von Neumann algebras, we refer the reader to [63, Chapter IV]. It is well-known that the isomorphism $\phi$ introduced in (5.1) is trace-preserving.

Before we proceed to the proof of the main result of this section (see Theorem 5.3 below), we establish several properties of the isomorphism $\phi$ introduced in (5.1) related to the generalised singular value functions and $\tau$-compact operators.

Proposition 5.1

Let $\phi$ be the spatial isomorphism from ${{\mathcal {M}}}$ onto ${\tilde{{{\mathcal {M}}}}}$ introduced in (5.1). Then, for any $X\in {{\mathcal {M}}}$, we have

• (i). $\mu (X) = \mu (\phi (X))$.

• (ii). $\mu (X) = \mu (X\otimes E_{00})$.

Proof

• (i). Since $\phi$ is an isometric, trace-preserving isomorphism from ${{\mathcal {M}}}$ onto ${\tilde{{{\mathcal {M}}}}}$, it follows from the definition of generalised singular value function (see Definition 2.1) that

  $$\begin{aligned} \mu (t; X)&= \inf \{\Vert XP\Vert _\infty : P \in {{\mathcal {P}}}({{\mathcal {M}}}), \tau (\mathbf{1 }-P) \le t\}\\&= \inf \{\Vert \phi (X) \phi (P) \Vert _\infty : P \in {{\mathcal {P}}}({{\mathcal {M}}}), \tau (\mathbf{1 }-P) \le t\} \\&= \inf \{\Vert \phi (X) {\tilde{P}} \Vert _\infty : {\tilde{P}} \in {{\mathcal {P}}}({\tilde{{{\mathcal {M}}}}}), {\tilde{\tau }}(\mathbf{1 }-{\tilde{P}}) \le t\}=\mu (t; \phi (X)). \end{aligned}$$

• (ii). For every $t>0$, we have

  $$\begin{aligned} d_{|X\otimes E_{00}|}(t )&={\tilde{\tau }}(E^{|X\otimes E_{00}|}(t,\infty )) \\&= {\tilde{\tau }}(E^{|X|\otimes E_{00}}(t,\infty )) \\&={\tilde{\tau }}(E^{|X|}(t,\infty )\otimes E_{00}) \\&=\tau (E^{|X|}(t,\infty ))= d_{|X|}(t). \end{aligned}$$

  Thus, by (2.1), we have $\mu (t; X\otimes E_{00}) = \mu (t; X)$.

$\square$

Proposition 5.2

Let $\phi$ be the spatial isomorphism from ${{\mathcal {M}}}$ onto ${\tilde{{{\mathcal {M}}}}}$ as in (5.1). We have that

• (i). ${{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}) = \phi ({{\mathcal {C}}}_0({{\mathcal {M}}},\tau ))$;

• (ii). If $K\otimes E_{00} \in {{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})$, then $K \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$;

• (iii). $({{\mathcal {M}}}\otimes {{\mathcal {L}}})\cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})= \{0\}.$

Proof

Part (i) immediately follows from Proposition 5.1.

(ii). Suppose that $K\otimes E_{00} \in {{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})$. On one hand, part (i) guarantees that $\phi ^{-1} (K\otimes E_{00} )\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. On the other hand, by Proposition 5.1, we have that $\mu (\phi ^{-1}(K\otimes E_{00}))= \mu (K)$. Hence, we conclude that $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$.

(iii). Let ${{\mathcal {J}}}({\tilde{{{\mathcal {M}}}}})$ be the norm closure of the linear space of all finite projections of ${\tilde{{{\mathcal {M}}}}}$. Since ${{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}) \subset {{\mathcal {J}}}({\tilde{{{\mathcal {M}}}}})$ (see Sect. 2), it follows from [44, Lemma 12 (b)] that $({{\mathcal {M}}}\otimes {{\mathcal {L}}})\cap {{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})=\{0\}$. $\square$

The lifting technique used in [44] (see also [21, 37]) and the already proven abelian case play crucial roles in proving Theorem 5.3. However, we can simplify the proof since the condition that ${{\mathcal {A}}}$ contains the center of ${{\mathcal {M}}}$ imposed in [44, Theorem 4] is not required in Theorem 4.2.

Theorem 5.3

Let ${{\mathcal {A}}}$ be a properly infinite von Neumann subalgebra of ${{\mathcal {M}}}$. For every derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, there exists $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\cap K_\delta$ such that $\delta =\delta _T$ on ${{\mathcal {A}}}$.

Proof

Let ${\tilde{\delta }} = \phi \circ \delta \circ \phi ^{-1}$, where $\phi$ is a spatial isomorphism as in (5.1). Clearly, ${\tilde{\delta }}$ is also a derivation, and, by Proposition 5.2, we have that

$$\begin{aligned} {\tilde{\delta }}: {{\tilde{{{\mathcal {A}}}}}} \rightarrow \phi ( {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )) = {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}). \end{aligned}$$

Let us define the following von Neumann algebras:

$$\begin{aligned} {\tilde{{{\mathcal {A}}}}}_1 = \mathbf{1 }\otimes {{\mathcal {L}}}, \quad {{\mathcal {A}}}_1=\phi ^{-1}({\tilde{{{\mathcal {A}}}}}_1), \quad {\tilde{{{\mathcal {A}}}}}_2 ={{\mathcal {A}}}\otimes {{\mathcal {L}}}\quad \text{ and } \quad {\tilde{{{\mathcal {A}}}}}_3 = {{\mathcal {A}}}_1\otimes {{\mathcal {D}}}. \end{aligned}$$

By Proposition 5.2 (iii) and [62, Chapter IV, Theorem 5.9 and Corollary 5.10], we have

$$\begin{aligned} {\tilde{{{\mathcal {A}}}}}_1' \cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})&= (\mathbf{1 }\otimes {{\mathcal {L}}})' \cap (({{\mathcal {M}}}\otimes B(H_0) \cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})) \nonumber \\&= ({{\mathcal {M}}}\otimes {{\mathcal {L}}}')\cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}) \nonumber \\&= ({{\mathcal {M}}}\otimes {{\mathcal {L}}})\cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}) \nonumber \\&=\{0\}. \end{aligned}$$

(5.2)

Since the isomorphism $\phi$ is spatial, we infer that

$$\begin{aligned} {{\mathcal {A}}}'_1\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )= \phi ^{-1} ({\tilde{{{\mathcal {A}}}}}_1') \cap {{\mathcal {C}}}_0 ({{\mathcal {M}}},\tau ) =\phi ^{-1} \left( {\tilde{{{\mathcal {A}}}}}_1' \cap {{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})\right) =\{0\}. \end{aligned}$$

(5.3)

We now study derivation ${\tilde{\delta }}$ on each of the algebras ${\tilde{{{\mathcal {A}}}}}_j$, $j=1,2,3$, separately.

Since ${\tilde{{{\mathcal {A}}}}}_1$ is abelian, Theorem 4.2 applied to the derivation ${\tilde{\delta }}|_{{\tilde{{{\mathcal {A}}}}}_1}$ guarantees the existence of $T_1 \in {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})$ such that

$$\begin{aligned} {\tilde{\delta }}_1 : = {\tilde{\delta }} -\delta _{T_1} \end{aligned}$$

vanishes on ${\tilde{{{\mathcal {A}}}}}_1$. Moreover, $T_1 \in {\overline{co}}^{wo}\{ U {\tilde{\delta }}(U^*)\mid U\in {{\mathcal {U}}}({\tilde{{{\mathcal {A}}}}}_1)\}$.

Note that ${\tilde{{{\mathcal {A}}}}}_2 ={{\mathcal {A}}}\otimes {{\mathcal {L}}}\subset {{\mathcal {M}}}\otimes {{\mathcal {L}}}\subset \mathbf{1 }'\otimes {{\mathcal {L}}}= {\tilde{{{\mathcal {A}}}}}_1'$. For any $A_1\in {\tilde{A}}_1$ and $A_2\in {\tilde{A}}_2$, we have

$$\begin{aligned} A_1{\tilde{\delta }}_1(A_2)= {\tilde{\delta }}_1(A_1A_2) = {\tilde{\delta }}_1(A_2A_1) = {\tilde{\delta }}_1(A_2) A_1, \end{aligned}$$

that is, ${\tilde{\delta }}_1(A_2)\in {\tilde{{{\mathcal {A}}}}}_1'$. Therefore, it follows from (5.2) that

$$\begin{aligned} {\tilde{\delta }}_1({\tilde{{{\mathcal {A}}}}}_2) \subset {\tilde{{{\mathcal {A}}}}}_1' \cap {{\mathcal {C}}}_0 ({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})= \{0\}, \end{aligned}$$

which implies that the derivation ${\tilde{\delta }}_1$ also vanishes on ${\tilde{{{\mathcal {A}}}}}_2$.

Next, we consider ${\tilde{\delta }}_1$ on the algebra ${\tilde{{{\mathcal {A}}}}}_3$. Since ${\tilde{{{\mathcal {A}}}}}_1$ is abelian, it follows that ${{\mathcal {A}}}_1$ is also abelian and therefore, ${\tilde{{{\mathcal {A}}}}}_3$ is also abelian. Thus, we can apply Theorem 4.2 to the derivation ${\tilde{\delta }}_1|_{{\tilde{{{\mathcal {A}}}}}_3}$ to infer that there is a $T_2\in {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})$ such that ${\tilde{\delta }}_1=\delta _{ T_2}$ on ${\tilde{{{\mathcal {A}}}}}_3$. We claim that $T_2 =0$, that is, ${\tilde{\delta }}_1$ vanishes on ${\tilde{{{\mathcal {A}}}}}_3$.

Since ${{\mathcal {A}}}_1 \otimes \mathbf{1 } \subset {{\mathcal {A}}}\otimes \mathbf{1 } \subset {{\mathcal {A}}}\otimes {{\mathcal {L}}}={\tilde{{{\mathcal {A}}}}}_2$, ${{\mathcal {A}}}_1 \otimes \mathbf{1 } \subset {\tilde{{{\mathcal {A}}}}}_3$ and ${\tilde{\delta }}_1$ vanishes on ${\tilde{{{\mathcal {A}}}}}_2$, we have $\delta _{ T_2}$ vanishes on ${{\mathcal {A}}}_1\otimes \mathbf{1 }$, i.e.,

$$\begin{aligned} T_2 \in ({{\mathcal {A}}}_1\otimes \mathbf{1 })' \cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})=({{\mathcal {A}}}_1'\otimes B(H_0)) \cap {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}). \end{aligned}$$

Hence, for all $i,j\in {\mathbb {Z}}$, we have that $(T_2)_{ij}\in {{\mathcal {A}}}_1'$ and

$$\begin{aligned} (T_2)_{ij}\otimes E_{00} = (\mathbf{1 }\otimes E_{0i})T_2(\mathbf{1 }\otimes E_{j0})\in {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }}). \end{aligned}$$

By Proposition 5.2 (ii), the latter condition implies that $(T_2)_{ij}\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and therefore, $(T_2)_{ij} \in {{\mathcal {A}}}'_1 \cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ for all $i,j\in {\mathbb {Z}}$. Appealing to (5.3), we conclude that $(T_2)_{ij}=0$ for all $i,j\in {\mathbb {Z}}$, so $T_2=0$. Thus, the derivation ${\tilde{\delta }}_1$ vanishes on ${\tilde{{{\mathcal {A}}}}}_3$. In particular, ${\tilde{\delta }}_1$ vanishes on $\mathbf{1 }\otimes {{\mathcal {D}}}$.

Finally, we claim that ${\tilde{\delta }}_1$ vanishes on ${\tilde{{{\mathcal {A}}}}}$, which would imply that ${\tilde{\delta }} = \delta _{T_1}$. Since ${{\mathcal {L}}}$ and ${{\mathcal {D}}}$ generate $B(H_0)$ in weak$^*$ operator topology, we have ${\tilde{{{\mathcal {A}}}}}_2 ={{\mathcal {A}}}\otimes {{\mathcal {L}}}$ and $\mathbf{1 }\otimes {{\mathcal {D}}}$ generate ${\tilde{{{\mathcal {A}}}}}$ in weak$^*$ operator topology. Since ${\tilde{\delta }}$ is weak$^*$ topology continuous (see [37, Lemma 1.3]), it follows that ${\tilde{\delta }}_1 ={\tilde{\delta }} -\delta _{ T_1} =0$, i.e., ${\tilde{\delta }} = \delta _{T_1}$ on ${\tilde{{{\mathcal {A}}}}}$. Then, for every $X\in {\tilde{{{\mathcal {A}}}}}$, we have

$$\begin{aligned} \phi (\delta (\phi ^{-1} (X))) ={\tilde{\delta }}(X) = \delta _ {T_1}( X) = XT_1- T_1X \end{aligned}$$

and therefore

$$\begin{aligned} \delta (\phi ^{-1} (X)) =\phi ^{-1}(XT_1- T_1X)=\phi ^{-1}(X)\phi ^{-1}(T_1)- \phi ^{-1}(T_1)\phi ^{-1}(X). \end{aligned}$$

(5.4)

Since $\phi$ is a isomorphism from ${{\mathcal {A}}}$ onto ${\tilde{{{\mathcal {A}}}}}$, (5.4) implies that $\delta (Y) =\delta _ {\phi ^{-1}(T_1)} (Y)$ for every $Y\in {{\mathcal {A}}}$. Since $T_1\in {{\mathcal {C}}}_0({\tilde{{{\mathcal {M}}}}},{\tilde{\tau }})\cap {\overline{co}}^{wo}\{ U {\tilde{\delta }}(U^*)\mid U\in {{\mathcal {U}}}({\tilde{{{\mathcal {A}}}}}_1)\}$, we have that $\phi ^{-1}(T_1)\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and $\phi ^{-1}(T_1)\in {\overline{co}}^{wo}\{ U \delta (U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}}_1)\}\subset K_\delta$, which completes the proof. $\square$

6 The Type I Case

Let ${{\mathcal {M}}}$ be a semifinite von Neumann algebra equipped with a faithful semifinite normal trace $\tau$. Derivations from a type I von Neumann subalgebra into $K({{\mathcal {H}}})$ and ${{\mathcal {J}}}({{\mathcal {M}}})$ are studied in [37] (see also [21, Section 10]) and [55], respectively. In what follows, we consider the case of derivations with values in ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Even though ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and ${{\mathcal {J}}}({{\mathcal {M}}})$ are similar in many respects and ${{\mathcal {J}}}({{\mathcal {M}}}) = {{\mathcal {C}}}_0({{\mathcal {M}}},{{\,\mathrm{\text{ Tr }}\,}}) =K({{\mathcal {H}}})$ when ${{\mathcal {M}}}=B({{\mathcal {H}}})$ and ${{\,\mathrm{\text{ Tr }}\,}}$ is the standard trace on $B({{\mathcal {H}}})$, the so-called locally compatible condition imposed in [55, Theorem 1.1] is redundant in the present setting. That is, we can consider the case when ${{\mathcal {A}}}$ is an arbitrary type I von Neumann subalgebra of ${{\mathcal {M}}}$. Before we proceed to the proof for the type I case, we need the following proposition.

Proposition 6.1

Let ${{\mathcal {A}}}$ be a type $I_n$ von Neumann subalgebra of ${{\mathcal {M}}}$, $n\in {\mathbb {N}}$. Then, every derivation $\delta$ from ${{\mathcal {A}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner, i.e., $\delta =\delta _{E}$ for some $E\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Moreover, $E\in co\{U\delta (U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\}$.

Proof

By [59, Theorem 2.3.3], we have ${{\mathcal {A}}}=M_n \otimes {{\mathcal {Z}}}({{\mathcal {A}}})$, where $M_n$ stands for the algebra of all $n\times n$ matrices. For the sake of convenience, we denote ${\mathcal {A}}=M_n\otimes {{\mathcal {Z}}}({\mathcal {A}})$ by $M_n({{\mathcal {Z}}}({\mathcal {A}}))$, and $E_{ij}\otimes \mathbf{1}_{{\mathcal {A}}}$ by $B_{ij}$, where $E_{ij}$ is the standard matrix units of $M_n$. In particular, every $A\in {\mathcal {A}}$ is in the form of $\sum _{i,j=1}^n A_{ij}B_{ij}$, $A_{ij} \in {{\mathcal {Z}}}({\mathcal {A}})$.

We define

$$\begin{aligned} D_1=\sum _{i=1}^n B_{i1}\delta (B_{1i}). \end{aligned}$$

Since every $\delta (B_{1i})$ is $\tau$-compact, it follows that $D_1$ is a $\tau$-compact operator.

Equality $\delta (\mathbf{1}_{{\mathcal {A}}})=0$ together with the Leibniz rule implies that

$$\begin{aligned} D_1&=\sum _{i=1}^n \Big (\delta (B_{i1} B_{1i} )-\delta (B_{i1})B_{1i} \Big )=\sum _{i=1}^n \Big (\delta (B_{ii})-\delta (B_{i1})B_{1i}\Big )\nonumber \\&=\delta (\mathbf{1}_{{\mathcal {A}}})-\sum _{i=1}^n \delta (B_{i1})B_{1i} =-\sum _{i=1}^n \delta (B_{i1})B_{1i} . \end{aligned}$$

(6.1)

Then, for every $k,l=1,\dots ,n$ we have

$$\begin{aligned} {[}B_{kl}, D_1]&= B_{kl} D_1 - D_1 B_{kl}{\mathop {=}\limits ^{(6.1)}} B_{kl}\sum _{i=1}^n B_{i1} \delta (B_{1i}) +\Big (\sum _{i=1}^n \delta (B_{i1})B_{1i}\Big ) B_{kl}\nonumber \\&=B_{k1}\delta (B_{1l})+\delta (B_{k1})B_{1l} =\delta (B_{k1}B_{1l})=\delta (B_{kl}). \end{aligned}$$

(6.2)

Now, consider $X=\sum _{i,j=1}^n X_{ij}B_{ij} \in {{\mathcal {A}}}$, $X_{ij}\in {{\mathcal {Z}}}({{\mathcal {A}}})$. Since $\sum _{k=1}^n X_{ij} B_{kk} \in {{\mathcal {Z}}}({{\mathcal {A}}})$, we have that $\delta \big (\sum _{k=1}^n X_{ij} B_{kk} \big )=0$ (see Remark 4.4). Hence, using the Leibniz rule, we write

$$\begin{aligned} \delta (X)&=\sum _{i,j=1}^n\delta ( X_{ij} B_{ij} )=\sum _{i,j=1}^n\delta \Big ((\sum _{k=1}^n X_{ij} B_{kk}) B_{ij} \Big )\\&=\sum _{i,j=1}^n\delta \big (\sum _{k=1}^n X_{ij} B_{kk} \big ) B_{ij} +\sum _{i,j=1}^n\big (\sum _{k=1}^n X_{ij} B_{kk} \big )\delta (B_{ij} )\\&=\sum _{i,j=1}^n\big (\sum _{k=1}^n X_{ij} B_{kk} \big )\delta (B_{ij}). \end{aligned}$$

Therefore, referring to (6.2), we obtain that

$$\begin{aligned} \delta (X)=\sum _{i,j=1}^n\big (\sum _{k=1}^n X_{ij} B_{kk} \big ) [B_{ij} ,D_1]. \end{aligned}$$

Since $\sum _{k=1}^n X_{ij} B_{kk} \in {{\mathcal {Z}}}({{\mathcal {A}}})$ and $\delta ({{\mathcal {Z}}}({{\mathcal {A}}}))=0$, it follows from the definition of $D_1$ that $\sum _{k=1}^n X_{ij} B_{kk}$ commutes with $D_1$. Hence, we obtain that

$$\begin{aligned} \delta (X)=\sum _{i,j=1}^n\big ( \sum _{k=1}^n X_{ij} B_{kk} \big )[B_{ij},D_1]=\sum _{i,j=1}^n\big [X_{ij}B_{ij} ,D_1 ]=[X,D_1]. \end{aligned}$$

Arguing similarly, one can show that $D_j:=\sum _{i=1}^n B_{ij} \delta (B_{ji} )$ such that $\delta =\delta _{D_j}$ for every j. Define

$$\begin{aligned} E:=\frac{1}{n} \sum _{j=1}^n D_j= \frac{1}{n} \sum _{i,j} B_{ij} \delta (B_{ji} ). \end{aligned}$$

(6.3)

Then, $\delta =\frac{1}{n} \sum _{j=1}^n \delta _{D_j}=\delta _E$. To complete the proof, it suffices to show that $E\in co\{U\delta (U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\}$.

We denote by S the collection of all (possibly empty) subsets of $\{1,\cdots ,n\}$. There are $2^n$ sets in S. For $i\in \{1,\cdots ,n\}$ and $K\in S$, we set $e^i_K=1$ if $i\in K$ and $e^i_K=-1$ if $i\notin K$. Let

$$\begin{aligned} a_{ij}:=\sum _{K\in S} e^i_Ke^j_K. \end{aligned}$$

(6.4)

Clearly, $a_{ii}=\sum _K 1= 2^n$.

Let $i\ne j$. We denote by $S_1$ the subset of S, such that every $K\in S_1$ satisfies $K\supset \{i,j\}$ and denote by $S_2$ the subset of S such that every $K\in S_2$ satisfies that $K\cap \{i,j\}=\varnothing$. Clearly, there are $2^{n-2}$ sets in $S_1$ and $2^{n-2}$ sets in $S_2$. For every $K\in S_1\cup S_2$, we have $e^i_Ke^j_K=1$. Note that there are $2^{n-1}$ sets in $S\setminus (S_1\cup S_2)$ and $e^i_Ke^j_K=-1$ for every $K\in S\setminus (S_1\cup S_2)$. Hence, for $i\ne j$, we have

$$\begin{aligned} a_{ij}=\sum _{K\in S} e^i_Ke^j_K=\sum _{K\in S_1 \cup S_2} e^i_Ke^j_K+\sum _{K\in S\setminus (S_1\cup S_2)} e^i_Ke^j_K=2^{n-1}-2^{n-1}= 0. \end{aligned}$$

(6.5)

For $\sigma \in S_{(n)}$, the set of all permutations of $\{1,2,\cdots ,n\}$, and $K\in S$, we define a unitary operator

$$\begin{aligned} U_\sigma ^K :=\sum _{i=1}^n e^i_K B_{i,\sigma (i)} . \end{aligned}$$

Then, by (6.4) and (6.5), we have

$$\begin{aligned} \sum _{\sigma \in S_{(n)}} \sum _{K\in S} U^K_\sigma \delta ((U^K_\sigma )^* )&=\sum _{\sigma \in S_{(n)}} \sum _{K\in S} \sum _{i,j} e^{i}_K e^j_K B_{j,\sigma (j)} \delta (B_{\sigma (i),i} ) \\&=\sum _{\sigma \in S_{(n)}} \sum _{i,j} B_{j,\sigma (j)} \delta (B_{\sigma (i),i} ) \sum _{K\in S} e^{i}_K e^j_K\\&=\sum _{\sigma \in S_{(n)}} \sum _{i}B_{i,\sigma (i)} \delta (B_{\sigma (i),i} ) a_{ii} \\&= 2^n \sum _{i}\sum _{\sigma \in S_{(n)}} B_{i,\sigma (i)} \delta (B_{\sigma (i),i} ) . \end{aligned}$$

For every i, j, there are $(n-1)!$ permutations taking i to j. Then, we obtain that

$$\begin{aligned} \sum _{\sigma \in S_{(n)}} \sum _{K\in S} U^K_\sigma \delta ((U^K_\sigma )^*) =2^n (n-1)!\sum _{i,j}B_{ij} \delta (B_{ji}) =2^n\frac{ n!}{n}\sum _{i,j}B_{ij} \delta (B_{ji}) {\mathop {=}\limits ^{(6.3)}}2^n n! E, \end{aligned}$$

which implies that $E\in co\{U\delta (U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\}$. $\square$

The following theorem is the main result of this section, which is a semifinite version of the so-called Johnson-Parrott theorem [37] (see also [21, Chapter 10]). Another semifinite version of the Johnson-Parrott theorem (see [55]) shows that derivations from a type I von Neumann subalgebra of ${{\mathcal {M}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$, the ideal of all compact operators in ${{\mathcal {M}}}$, are not necessarily inner. However, in the following theorem, we show that derivations from an arbitrary type I von Neumann subalgebra of ${{\mathcal {M}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ are necessarily inner.

Theorem 6.2

If ${{\mathcal {A}}}$ is a type I von Neumann subalgebra of ${{\mathcal {M}}}$, then for every derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, there exists $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\cap K_\delta$ such that $\delta =\delta _K$.

Proof

Since ${{\mathcal {A}}}$ is a type I von Neumann algebra, there exists a central partition of unity $\{Z_n: n\in {\mathbb {N}}\}$ such that $Z_n {{\mathcal {A}}}$ is of type $I_{n}$ and $Z_0{{\mathcal {A}}}$ is properly infinite. Recall that we may always assume that $\delta |_{{{\mathcal {Z}}}({{\mathcal {A}}})} =0$ (see Remark 4.4). We have $\delta (Z_n{{\mathcal {A}}})\subset Z_n {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_n$ for all $n\ge 0$. Since for $n\ge 1$, the algebra $Z_n{{\mathcal {A}}}$ is of type $I_n$, it follows from Proposition 6.1 that $\delta |_{Z_n{{\mathcal {A}}}}=\delta _{K_n}$ for some $K_n\in Z_n {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_n\cap Z_n K_\delta$. In addition, by Theorem 5.3, there exists $K_0\in Z_0{{\mathcal {C}}}_0({{\mathcal {M}}},\tau )Z_0 \cap Z_0 K_\delta$ such that $\delta |_{Z_0{{\mathcal {A}}}}=\delta _{K_0}$. Set $K=\sum _{n=0} ^\infty Z_n K_n$. Appealing to Theorem 3.9, we conclude that that $\delta =\delta _K$ and $K \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\cap K_\delta$. $\square$

7 The Type $II_1$ Case

Recall that ${{\mathcal {M}}}$ is a semifinite von Neumann algebra with a faithful normal semifinite trace $\tau$. Let ${{\mathcal {A}}}$ be a von Neumann subalgebra of ${{\mathcal {M}}}$ and let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation. As we showed in Theorems 5.3 and  6.2, the derivation $\delta$ is inner provided that ${{\mathcal {A}}}$ is properly infinite or of type I. Hence, by Remark 3.4, to complete the proof of Theorem 1.2, it remains to consider the case when ${{\mathcal {A}}}$ is of type $II_1$. We cover this remaining case in the present section.

The special case when ${{\mathcal {M}}}=B({{\mathcal {H}}})$ (in this case, ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is just K(H)) was first resolved by Popa (see [54, Theorem II]) by transforming the noncommutative framework of the problem into a commutative one. In [55], Popa and Rădulescu extended the results in [37, 54] to a semifinite case when $\delta$ takes values from ${{\mathcal {J}}}({{\mathcal {M}}})$, the ideal of all compact operators in ${{\mathcal {M}}}$ (see also [32, Section 5] for the case when the smaller algebra is a type $II_1$ factor, and [44] for the case when the smaller algebra is abelian or properly infinite which contains the center of ${{\mathcal {M}}}$).

In the setting of the present section, we consider a derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, where ${{\mathcal {A}}}$ is a type $II_1$ algebra. Since ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {J}}}({{\mathcal {M}}})$, the main result of [55] guarantees that there exists $T\in {{\mathcal {J}}}({{\mathcal {M}}})$ such that $\delta =\delta _T$. Hence, to prove that $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner, it is sufficient to show that there exists $T'\in {{\mathcal {A}}}'$ such that $T-T' \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$.

7.1 Some preliminaries

Let ${{\mathcal {M}}_\tau }:= \{X\in {{\mathcal {M}}}\mid \tau (X^* X) <\infty \}$ be the Hilbert-Schmidt class ideal in ${{\mathcal {M}}}$ equipped with the norm $\Vert X\Vert _\tau = \tau (X^* X)^{\frac{1}{2}}$, $X\in {{\mathcal {M}}_\tau }$. Let ${{\mathcal {H}}}_\tau$ be the Hilbert space completion of ${{\mathcal {M}}_\tau }$ in the norm $\left\| \cdot \right\| _\tau$, that is, ${{\mathcal {H}}}_\tau = L_2({{\mathcal {M}}},\tau )$. ${{\mathcal {M}}}$ is always regarded in its standard representation, acting on ${{\mathcal {H}}}_\tau$ by left multiplication.

In what follows, we introduce norms and on ${{\mathcal {M}}}$. The norms and play similar roles in this paper as the uniform and usual essential norms do in [37] and [54] (see also [21, Chapter 10]).

By the well-known Holmstedt formula (see e.g. [34, Theorem 4.1]), $\left\| \cdot \right\| _{L_2+L_\infty }$ defined by $\left\| f\right\| _{L_2+L_\infty }=(\int _0^{1}\mu (t;f)^2 dt)^{1/2}$, $f\in L_2(0,\infty ) +L_\infty (0,\infty )$, is a complete norm on $L_2(0,\infty ) +L_\infty (0,\infty )$. It follows immediately from the definition of the norm $\left\| \cdot \right\| _{L_2+L_\infty }$ that $(L_2+L_\infty )(0,\infty )$ equipped with the norm $\left\| \cdot \right\| _{L_2+L_\infty }$ is a strongly symmetric space. Hence, $(L_2+L_\infty )({{\mathcal {M}}},\tau )$ is a strongly symmetric operator space equipped with norm $\Vert \cdot \Vert _{L_2+L_\infty }$ defined by $\Vert T\Vert _{L_2+L_\infty }=(\int _0^{1}\mu (t;T)^2 dt)^{1/2}$, $T\in (L_2+L_\infty )({{\mathcal {M}}},\tau )$ (see e.g. [28, 30, 45]).

Note that the definition of below is rather different from that of the uniform norms introduced in [37, 54] and [55, 2.3].

Definition 7.1

For every $T\in {{\mathcal {M}}}$, we define . It is clear that .

Proposition 7.2

If $T_1, T_2,T\in {{\mathcal {M}}}$, then and .

Lemma 7.3

Let $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and $\{E_n\}$ be a sequence of mutually orthogonal projections in ${{\mathcal {M}}}$. Then, we have and .

Proof

By Proposition 7.2, we have . Therefore, it is sufficient to show that .

Since K is a $\tau$-compact operator, the projection $E^{|K|}(\varepsilon ,\infty )$ is $\tau$-finite for every $\varepsilon >0$. In particular, $KE^{|K|}(\varepsilon ,\infty ) \in {{\mathcal {F}}}({{\mathcal {M}}},\tau )$. Since , it follows that . Then, [28, Proposition 56] together with [27, Theorem 6.13 (iii)] implies that . By Proposition 7.2, we have . $\square$

Theorem 7.4

is inferior semicontinuous with respect to the weak operator topology, that is, if a net $\{T_i\}$ converges T in the weak operator topology, then .

Proof

By [47, Lemma 2.3.18], we may assume without loss of generality that ${{\mathcal {M}}}$ is atomless. We have that $\mu (T)^2 = \mu (|T|)^2 =\mu (|T|^2 )$ (see [47, Corollary 2.3.17 (d)]). Therefore, by [47, Lemma 3.3.2], we have that

$$\begin{aligned} \int _0^1 \mu (s;T)^2ds=\int _0^1 \mu (s;|T|^2)ds =\sup \{\Vert TP\Vert _\tau ^2: P\in {{\mathcal {P}}}({{\mathcal {M}}}),~\tau (P)\le 1\}. \end{aligned}$$

(7.1)

and, similarly,

$$\begin{aligned} \int _0^1 \mu (s;T_i)^2ds=\int _0^1 \mu (s;|T_i|^2)ds =\sup \{\Vert T_iP\Vert _\tau ^2: P\in {{\mathcal {P}}}({{\mathcal {M}}}),~\tau (P)\le 1\}. \end{aligned}$$

(7.2)

Let $P\in {{\mathcal {P}}}({{\mathcal {M}}})$ be such that $\tau (P)\le 1$. Since $|\tau (PT^* T_i P)|\le \Vert T_iP\Vert _\tau \Vert TP\Vert _\tau$, it follows that $\Vert TP\Vert _\tau ^2 = \tau ( PT^*TP)=\lim _i |\tau (PT^* T_i P)|\le \limsup _i\Vert T_iP\Vert _\tau \Vert TP\Vert _\tau$. Hence, we have

which together with (7.1) implies that . $\square$

In [55, 2.6], an essential norm was defined in terms of the compact ideal in a semifinite von Neumann algebra. Below, we introduce the essential norm with respect to the $\tau$-compact ideal.

Definition 7.5

For $T\in {{\mathcal {M}}}$, we define .

The norm can be described in terms of the singular value function.

Proposition 7.6

for every $T\in {{\mathcal {M}}}$.

Proof

We may assume that $\mu (\infty ;T) =\varepsilon$ for some $\varepsilon >0$. For any $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and $\Delta >0$, there exists $t_0>0$ such that $\mu (t;T)-\mu (t;K)\ge \varepsilon -\Delta$ for every $t>t_0$. Using [23, Theorem 3.4] (see also [28]), we obtain that

$$\begin{aligned} \varepsilon -\Delta \prec \prec \mu (T ) -\mu ( K) \prec \prec \mu (T - K) \end{aligned}$$

for any $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ and $\Delta >0$. By Lemma 2.6, the latter condition guarantees that $\mu (T - K) \ge \varepsilon -\Delta$. Thus, .

Since K and $\Delta$ are arbitrary, it follows that .

To prove the converse inequality, assume that $\Delta > 0$ and choose $t>0$ such that $\mu (t;T)\le \varepsilon +\Delta$. By Lemma 2.4, $E^{|T|} (\varepsilon +\Delta ,\infty )$ is $\tau$-finite. In particular, $TE^{|T|} (\varepsilon +\Delta ,\infty )\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. It follows from the definition of that

Since $\Delta$ is arbitrary, we obtain that . $\square$

Let $T_1,T_2\in {{\mathcal {M}}}$ be two operators which are disjoint from the left and the right. The essential norm of $T_1+T_2$ with respect to ${{\mathcal {J}}}({{\mathcal {M}}})$ (see [55, Definition 2.6]) does not necessarily equal the maximum of the essential norms of $T_1$ and $T_2$ with respect to ${{\mathcal {J}}}({{\mathcal {M}}})$ (see [55, Section 2.7]). However, similar to the usual essential norm in $B({{\mathcal {H}}})$ (see e.g. [21, 37]), the essential norm with respect to ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ has the following property for disjointly supported operators.

Proposition 7.7

Let $T\in {{\mathcal {M}}}$ and let $P_1, P_2$ be mutually orthogonal projections in ${{\mathcal {M}}}$. We have

(7.3)

Proof

Without loss of generality, we assume that both $P_1TP_1$ and $P_2T P_2$ are not $\tau$-compact with $\varepsilon _1 := \mu (\infty ; P_1TP_1 ) \ge \mu (\infty ; P_2TP_2) =: \varepsilon _2$. By Lemma 2.4, for every $\Delta >0$, we have

$$\begin{aligned} \tau (E^{|P_1 T P_1 +P_2 T P_2|} (\varepsilon _1+\Delta ,\infty ))&=\tau (E^{|P_1 T P_1| } (\varepsilon _1+\Delta ,\infty ))+\tau (E^{|P_2 T P_2|} (\varepsilon _1+\Delta ,\infty ))\\&=M<\infty \end{aligned}$$

for some $M>0$. Hence, using again Lemma 2.4, we obtain that

$$\begin{aligned} \mu ( \infty ;P_1 T P_1 +P_2 T P_2) \le \mu ( M ;P_1 T P_1 +P_2 T P_2) \le \varepsilon _1+\Delta . \end{aligned}$$

Moreover, since $|P_1TP_1| \le |P_1 T P_1| +|P_2 T P_2| =|P_1 T P_1 +P_2 T P_2|$, it follows that

$$\begin{aligned} \varepsilon _1=\mu (\infty ;P_1TP_1)\le \mu (\infty ;P_1 T P_1 +P_2 T P_2). \end{aligned}$$

Since $\Delta$ is arbitrary, we conclude that $\mu (\infty ;P_1 T P_1 +P_2 T P_2)=\varepsilon _1$. The assertion now follows from Proposition 7.6. $\square$

Remark 7.8

Note that for any semifinite von Neumann algebra ${{\mathcal {A}}}$ there exist pairwise orthogonal central projections $P_i$ with $\sum _i P_i=1$ such that each ${{\mathcal {Z}}}({{\mathcal {A}}})_{ P_i}$ is countably decomposable. Hence, combining Theorem  3.9 together with Theorem 4.2, we may assume, without loss of generality, that the center ${{\mathcal {Z}}}({{\mathcal {A}}})$ of the von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ is countably decomposable. In particular, since ${{\mathcal {A}}}$ is of type $II_1$, we can always assume that ${{\mathcal {A}}}$ is a countably decomposable type $II_1$ von Neumann algebra (see e.g. [42, Corollary 8.2.9]).

7.2 Some continuity results

In this subsection, we study the continuity of derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Similar results with respect to the ideal ${{\mathcal {J}}}({{\mathcal {M}}})$ can be found in [55, Section 4]. In this part, unless otherwise stated, we always assume that algebra ${{\mathcal {A}}}$ is a countably decomposable type $II_1$ von Neumann subalgebra of ${{\mathcal {M}}}$ and therefore ${{\mathcal {A}}}$ has a normal faithful finite trace $\tau _{{\mathcal {A}}}$. For every $X\in {{\mathcal {A}}}$, we denote

$$\begin{aligned} \Vert X\Vert _2 := \tau _{{\mathcal {A}}}(X^*X)^{1/2}. \end{aligned}$$

Proposition 7.9

Let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation. Then, $\delta$ is continuous from the unit ball of ${{\mathcal {A}}}$ with the strong operator topology into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ equipped with the norm .

Proof

By Ringrose’s theorem [56], the mapping is continuous. Hence, denoting by $\Vert \delta \Vert$ the operator norm this mapping, we can assume that $\Vert \delta \Vert \le 1$.

We firstly prove that if $\{ P_n \}_{n\in {\mathbb {N}}}$ is a sequence of projections in ${{\mathcal {A}}}$ with $\tau _{{\mathcal {A}}}(P_n) \rightarrow 0$, then Suppose that does not converge to 0. Passing to a subsequence, if necessary, we may assume that for some $c >0$ for all n and that $\sum \tau _{{\mathcal {A}}}(P_n) <\infty$. Define $G_n := \vee _{k \ge n } P_k$. We have

$$\begin{aligned} \tau _{{\mathcal {A}}}( G_n ) \le \sum _{k\ge n } \tau _{{\mathcal {A}}}(P_k) \rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$. Denote by $S_{n,m}$ the support of $P_m G_n P_m$. It is clear that $S_{n,m} \le P_m$. Moreover, since $S_{n,m}=l(P_m G_n P_m)=r(P_m G_n P_m)$, it follows that $l(P_m G_n P_m)\le l(P_m G_n)\sim r(P_m G_n)\le G_n$, i.e. $S_{n,m}\preceq G_n$. Therefore, $\tau _{{\mathcal {A}}}(S_{n,m})\le \tau _{{\mathcal {A}}}(G_n) \rightarrow _n 0$ for each m. Since $\{G_n\}_n$ is decreasing, it follows that for every fixed m, the sequence $\{P_m G_n P_m\}_n$ is decreasing, and so, $\{S_{n,m}\}$ is decreasing, too. In particular, $S_{n,m}\downarrow _0$ as $n\rightarrow \infty$. Thus, $\{ P_m -S_{n,m} \}_n$ increases to $P_m$. Since $\delta$ is continuous in weak$^*$ operator topology (see [37, Lemma 1.3]), we obtain that $\{\delta ( P_m -S_{n,m} )\}$ is convergent to $\delta (P_m)$ in the weak$^*$ operator topology. By the inferior semicontinuity of the norm (see Theorem 7.4), it follows that for a fixed m, we can find a sufficiently large n such that

Thus, by induction, we can find an increasing sequence of integers $n_1,n_2,\cdots$ such that for every k, the projection $H_k := P_{n_k} - S_{n_{k+1},n_k}$ satisfies . These projections also satisfy $\tau _{{\mathcal {A}}}(H_k) \le \tau _{{\mathcal {A}}}(P_{n_k} ) \rightarrow _k 0$. Moreover, since $H_k \le P_{n_k}$ and $S_{n_{k+1},n_k}$ is the support of $P_{n_k}G _{n_{k+1}} P_{n_k}$, by the definition of $H_k$ we get

$$\begin{aligned} H_k G_{n_{k+1}} H_k = H_k P_{n_{k}} G_{n_{k+1}} P_{n_k} H_{k}= H_k P_{n_{k}} G_{n_{k+1}} P_{n_k} S_{n_{k+1},n_k} H_k =0 , \end{aligned}$$

which implies that $H_k G_{n_{k+1}} =0$. Recalling that $G_n = \vee _{k\ge n} P_k$, we conclude that $H_k H_{l} =0$ for every $l \ge k+1$, which means that $H_k$ are mutually orthogonal projections.

Denote by ${{\mathcal {B}}}$ the abelian von Neumann subalgebra of ${{\mathcal {A}}}$ generated by $\{H_k\}$. By considering $\delta$ as a derivation from ${{\mathcal {B}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, we can apply Theorem 4.2 to obtain the existence of $K\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, such that $\delta (H_k )=\delta _K(H_k)$ for $k\in {\mathbb {N}}$. On one hand, . On the other hand, since $H_n$ are mutually orthogonal projections, Lemma 7.3 implies that , which is a contradiction.

Now, we turn to the general case. It is well-known that $\Vert \cdot \Vert _2$ induces the strong operator topology on the unit ball of ${{\mathcal {A}}}$. Hence, it suffices to prove that if $\{X_n \}_n$ is a bounded sequence in ${{\mathcal {A}}}$ with $\Vert X_n\Vert _2 \rightarrow _n 0$ , then . Without loss of generality, we may assume that every element in $\{X_n\}$ is positive and $\Vert X_n\Vert _\infty \le 1$ (see e.g. a similar argument in the penultimate paragraph of the proof of [55, Proposition 4.1]).

Let $0\le X\le {\mathbf {1}}$ be arbitrary. Let $A_m = \cup _{i=1}^{2^{m-1}} ((2i-1)/2^m, 2i/2^m]$, $m\ge 1$. We define

$$\begin{aligned} k_m := \chi _{A_m}, ~m\ge 1. \end{aligned}$$

Note that for every $\lambda \in [0,1]$, we have $\lambda = \sum _{m\ge 1} 2^{-m} k_m(\lambda )$. By functional calculus, we have

$$\begin{aligned} X&=\int \lambda dE^X_\lambda =\int \sum _{m\ge 1} 2^{-m}k_m(\lambda ) dE^X_\lambda \\&=\sum _{m\ge 1} 2^{-m}\int k_m(\lambda )dE^X_\lambda =\sum _{m\ge 1} 2^{-m}E_{A_m}^X. \end{aligned}$$

Thus, for every $X_n$, we can write the dyadic decomposition

$$\begin{aligned} X_n : = \sum _{m\ge 1} 2^{-m} e^{n}_m, \end{aligned}$$

where $e^n_m:= E_{A_m}^{X_n}$.

Since $\Vert X_n\Vert _2 \rightarrow _n 0$, it follows that $\tau _{{\mathcal {A}}}(e^{n}_m) \rightarrow _n 0$ for each $m \ge 1$. Let $\varepsilon >0$ be fixed and choose $m_0 \ge 1$ such that $2^{- m_0}\le \frac{\varepsilon }{2}$. By the first part of the proof, there exists $n_0$ such that for every $n \ge n_0$, for any $m \le m_0$. Recall that $\Vert \delta \Vert \le 1$. For $n \ge n_0$, we infer that

which completes the proof. $\square$

Recall that $K_\delta ={\overline{co}}^{wo} \{ U \delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \}$. Proposition 7.9 immediately implies the following corollary.

Corollary 7.10

Given $\beta >0$, there exists $\alpha >0$ such that

for all $T\in K_\delta$ and $X\in {{\mathcal {A}}}$, $\Vert X\Vert _\infty \le 1$, $\Vert X\Vert _2 \le \alpha$.

Proof

By Proposition 7.9, there exists $\alpha > 0$ such that for every $X\in {{\mathcal {A}}}$ with $\Vert X \Vert _\infty \le 1$ and $\Vert X \Vert _2 <\alpha$. For a unitary element U in ${{\mathcal {A}}}$, we have $U\delta (U^* ) X = U \delta (U^*X ) -\delta (X)$ and $\Vert U^* X\Vert _2 = \Vert X \Vert _2$, which implies that

By taking convex combinations of $U \delta (U^*)$ and using the inferior semi-continuity of norm in the weak operator topology (see Theorem  7.4), we get for all $T\in K_\delta$. The symmetricity of the norm (see Proposition 7.2) implies that . $\square$

The following proposition is the main result of the present subsection, which is the key in the proof of Theorem 7.13 below.

Proposition 7.11

Let $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ be a derivation. If $T\in K_\delta$ is such that $\delta = \delta _T$ on ${{\mathcal {A}}}$, then $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$.

Proof

Since ${{\mathcal {A}}}$ is of type $II_1$, there exists a decreasing sequence of projections $\{E_n\}_{n\ge 0}$ in ${{\mathcal {A}}}$ with $E_0 =\mathbf{1}$, $E_{n+1 } \sim E_{n} -E_{n+1}$ for all $n \ge 0$ (see e.g. [42, Lemma 6.5.6]).

Using mathematical induction, we show that for all n. For $n = 0$, the assertion is trivial. Assume that for all $k\le n$ for some fixed $n\ge 0$. For every n, by [64, Chapter XIV, Lemma 2.1], there is a unitary element $U_n\in {{\mathcal {A}}}$ such that

$$\begin{aligned} U_n^* E_{n+1} U_n =E_n -E_{n+1} . \end{aligned}$$

(7.4)

Since $\delta =\delta _T$, it follows that that $U_n^* T U_n -T = \delta (U^*)U \in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. Therefore, by Proposition 7.6 and Definition 7.5, we have

(7.5)

Now, using now Proposition 7.7, we infer that

Therefore, , which concludes the induction argument.

Assume now that $T \notin {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, that is,

(7.6)

for all n.

Since $\vee _{k \ge n } E_k \downarrow _n 0$, we have $\tau _{{\mathcal {A}}}(E_n) \le \tau _{{\mathcal {A}}}(\vee _{k \ge n } E_k ) \rightarrow _n 0$. Since $T\in K_\delta$ and $\Vert E_n\Vert _2 \rightarrow 0$, Proposition 7.2 and Corollary 7.10 imply that

which is a contradiction to (7.6). Thus, $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, as required. $\square$

7.3 The proof for the main result: the type $II_1$ case

Before proceeding to the proof of Theorem 7.13, we prove the special case when ${{\mathcal {Z}}}({{\mathcal {M}}})$ is of countable type by using the auxiliary results obtained in Sect. 7.2 and [55, Section 7.4]. To prove the case for ${{\mathcal {J}}}({{\mathcal {M}}})$, several reductions are needed in [55, Section 7]. However, rather than repeating the proof in [55, Section 7], we use the main result of [55] in the proof of the following proposition, which makes our proof more efficient.

Proposition 7.12

If the center ${{\mathcal {Z}}}({{\mathcal {M}}})$ of ${{\mathcal {M}}}$ is countably decomposable, then every derivation $\delta$ from a type II$_1$ von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner. Moreover, the element $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ implementing $\delta$ lies in $K_\delta$.

Proof

It is proved in [55] that every derivation from a type $II_1$ von Neumann subalgebra ${{\mathcal {A}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ is implemented by some element in ${{\mathcal {J}}}({{\mathcal {M}}})$. Noticing that ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {J}}}({{\mathcal {M}}})$, we conclude that there exists an element $K\in {{\mathcal {J}}}({{\mathcal {M}}})$ such that $\delta =\delta _K$. Since ${{\mathcal {Z}}}({{\mathcal {M}}})$ is countably decomposable, by [55, Lemma 4.6] (note that this Lemma requires the condition that ${{\mathcal {Z}}}({{\mathcal {M}}})$ is countably decomposable), there is a ${\overline{T}}\in {\overline{co}}^{wo} \{ \delta (U) U^* \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}) \} =-K_\delta$ such that $\delta (\cdot ) =[{\overline{T}},\cdot ] = \delta _{-{\overline{T}}}(\cdot )$. Now, let $T=-{\overline{T}}$. Then, $T\in K_\delta$ with $\delta =\delta _T$. It follows from Proposition 7.11 that $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. $\square$

In the following theorem, we remove the condition that ${{\mathcal {Z}}}({{\mathcal {M}}})$ is countably decomposable imposed in Proposition 7.12, proving the main result of this section.

Theorem 7.13

Every derivation $\delta$ from a type II$_1$ von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is inner. Moreover, the element implementing $\delta$ lies in $K_\delta$.

Proof

Let $\{Z_i \in {{\mathcal {Z}}}({{\mathcal {M}}})\}$ be a net of projections increasing to $\mathbf{1}$ such that ${{\mathcal {Z}}}({{\mathcal {M}}}_{Z_i}) = {{\mathcal {Z}}}({{\mathcal {M}}})_{Z_i}$ is countably decomposable. Since ${{\mathcal {A}}}$ is assumed to be countably decomposable (see Remark 7.8), it follows that ${{\mathcal {A}}}_{Z_i}$ is also countably decomposable. Define $\delta _i :{{\mathcal {A}}}_{Z_i} \rightarrow Z_i {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) Z_i ={{\mathcal {C}}}_0({{\mathcal {M}}}_{Z_i },\tau )$ by $\delta _i(X Z_i ) = Z_i \delta (X)Z_i$ for every $X\in {{\mathcal {A}}}$. Since $Z_i \in {{\mathcal {Z}}}({{\mathcal {M}}})$, it follows from Lemma 3.2 that $\delta _i$ are well-defined derivations.

By Proposition 7.12, there exists $K_i \in {{\mathcal {C}}}_0({{\mathcal {M}}}_{Z_i},\tau )$ with $K_i \in K_{\delta _i}$ such that $\delta _i =\delta _{K_i}$ on ${{\mathcal {A}}}_{Z_i}$. Since ${{\mathcal {U}}}({{\mathcal {A}}}_{Z_i}) = {{\mathcal {U}}}({{\mathcal {A}}}) Z_i$ (see e.g. [41, Proposition 5.5.5]), it follows that

$$\begin{aligned} K_i \in K_{\delta _i}&= {\overline{co}}^{wo} \{U\delta _i(U^*)\mid U\in {{\mathcal {U}}}({{\mathcal {A}}}_{Z_i})\} ={\overline{co}}^{wo} \{UZ_i \delta _i(U^* Z_i )\mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} \\&= {\overline{co}}^{wo} \{U Z_i\delta (U^*) Z_i \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} =K_\delta Z_i. \end{aligned}$$

Hence, for every i, there exists $T_i\in K_\delta$ such that $K_i=Z_i T_i Z_i$.

Note that $K_\delta$ is compact in the weak operator topology (see [20, Chapter IX, Proposition 5.5]). Let $T\in K_\delta$ be a limit point of a subnet of $\{T_i\}_i$ in the weak operator topology. Without loss of generality, we assume that $T_i \rightarrow _{wo} T$. For every $X\in {{\mathcal {A}}}$, we have that

$$\begin{aligned} Z_i \delta (X) Z_i = \delta _{K_i}(X) = \delta _{Z_iT_i Z_i}(X). \end{aligned}$$

On one hand, since $Z_i\uparrow \mathbf{1}$, it follows that $Z_i\delta (X)Z_i =Z_i\delta (X) \rightarrow _{so} \delta (X)$ and therefore a fortiori $Z_i\delta (X)Z_i \rightarrow _{wo} \delta (X)$. On the other hand, since $\Vert U\delta (U^*)\Vert _\infty \le \Vert \delta \Vert _{({{\mathcal {A}}},\Vert \cdot \Vert _\infty ) \rightarrow ({{\mathcal {C}}}_0({{\mathcal {M}}},\tau ),\Vert \cdot \Vert _\infty ) }<\infty$ (see [56, Theorem 2]) and $T_i \in K_\delta$, it follows that $\Vert T_i\Vert _\infty \le \Vert \delta \Vert _{({{\mathcal {A}}},\Vert \cdot \Vert _\infty ) \rightarrow ({{\mathcal {C}}}_0({{\mathcal {M}}},\tau ),\Vert \cdot \Vert _\infty ) }<\infty$ for every i. Hence, $Z_iT_iZ_i\rightarrow _{wo} T$. Combining these two convergences, we conclude that

$$\begin{aligned} \delta (X) =wo\text{- }\lim _i Z_i \delta (X) Z_i = wo\text{- }\lim _i \delta _{Z_iT_i Z_i}(X)= \delta _T(X), \end{aligned}$$

that is, $\delta =\delta _T$ on ${{\mathcal {A}}}$. Since $T\in K_\delta$, it follows from Proposition 7.11 that $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$.

$\square$

8 Conclusions and Applications

The following theorem is the main result of the present paper, which shows that for an arbitrary von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$, every derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is necessarily inner. We note that this result is in contrast to [55, Theorem 1.2] (see also [59, Example 4.1.8] and [10] for derivations acting on $C^*$-subalgebras). We do not impose any additional assumptions of the type I summand of the algebra ${{\mathcal {A}}}$.

Theorem 8.1

Every derivation $\delta$ from a von Neumann subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$ into the ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ of all $\tau$-compact operators is inner.

Proof

By Theorem 4.2, there exists $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ such that $\delta |_{{{\mathcal {Z}}}({{\mathcal {A}}})} =\delta _T$. Replacing $\delta$ with $\delta -\delta _T$, we can assume that $\delta$ vanishes on ${{\mathcal {Z}}}({{\mathcal {A}}})$. By Remark 3.4, it suffices to prove the assertion in the case when ${{\mathcal {A}}}$ is of type I, type $II_1$ and properly infinite, separately. Hence, appealing to Theorem 6.2, Theorem 7.13 and Theorem 5.3, we conclude the proof. $\square$

In the particular case when ${{\mathcal {M}}}=B({{\mathcal {H}}})$ and $\tau$ is the standard trace, our result recovers the results proved by Johnson and Parrott [37], and by Popa [54]. Furthermore, in the case of an arbitrary von Neumann algebra ${{\mathcal {M}}}$ equipped with a faithful normal finite trace $\tau$, we have that ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) ={{\mathcal {M}}}$ (see e.g. [47, Page 64]), and therefore, Theorem 8.1 guarantees that any derivation $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}$ is inner if ${{\mathcal {M}}}$ is equipped with a faithful normal finite trace. In the following corollary, we extend this result to a general finite von Neumann algebra ${{\mathcal {M}}}$, recovering the main result of [15, Section 5] using completely different approach.

Corollary 8.2

Every derivation $\delta$ from a von Neumann subalgebra ${{\mathcal {A}}}$ of a finite von Neumann algebra ${{\mathcal {M}}}$ into ${{\mathcal {M}}}$ is inner. Moreover, the element $K\in {{\mathcal {M}}}$ implementing $\delta$ can be chosen from $K_\delta$.

Proof

Since ${{\mathcal {M}}}$ is finite, it follows that there is a net $\{P_i\}$ of projections in ${{\mathcal {Z}}}({{\mathcal {M}}})$ with $P_i\uparrow \mathbf{1}$ such that ${{\mathcal {M}}}_{P_i}$ is countably decomposable (see e.g. [42, Corollary 8.2.9] and [51, Theorem 1.3.6]). Hence, ${{\mathcal {M}}}_{P_i}$ has a faithful normal finite trace $\tau _i$ (see e.g. [51, Theorem 1.3.6]), that is, ${{\mathcal {M}}}_{P_i} ={{\mathcal {C}}}_0({{\mathcal {M}}}_{P_i},\tau _i)$. Therefore, by Theorem 8.1, the derivation $\delta _i:{{\mathcal {A}}}_{P_i}\rightarrow {{\mathcal {M}}}_{P_i}$ defined by $\delta _i(X P_i) = \delta (X) P_i$ is inner, that is, there exists $T_i \in {{\mathcal {M}}}_{P_i}$ such that $\delta _i= \delta _{T_i}$ on ${{\mathcal {A}}}_ {P_i}$. By Proposition 3.10, there exists

$$\begin{aligned} K_i \in K_{\delta _i}&= {\overline{co}}^{wo}\{U\delta _i(U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}}_{P_i})\} ={\overline{co}}^{wo}\{U P_i\delta _i (U^*P_i ) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} \\&={\overline{co}}^{wo}\{U\delta (U )P_i \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\} = K_\delta P_i \end{aligned}$$

such that $\delta _i=\delta _{K_i}$ on ${{\mathcal {A}}}_{P_i}$ and there is a $K'_i\in K_\delta$ such that $K_i= K'_iP_i$. Since $K_\delta$ is compact in the weak operator topology (see [20, Chapter IX, Proposition 5.5]), there is a limit point $K\in K_\delta$ of a subnet of $\{K'_i \}$ in the weak operator topology. Without loss of generality, we denote that $K_i'\rightarrow _{wo} K$. We have

$$\begin{aligned} \delta (X) P_i= \delta _i(X) =\delta _{K_i}(X) = \delta _{K'_iP_i}(X ) \end{aligned}$$

for every $X\in {{\mathcal {A}}}$. Since $\{P_i\}$ converges strongly to the identity, it follows that $\delta =\delta _K$ on ${{\mathcal {A}}}$. $\square$

We conclude our paper with applications of Theorem  8.1 to derivations with values in a class of ideals of ${{\mathcal {M}}}$. We characterize a class of ideals ${{\mathcal {E}}}$ of ${{\mathcal {M}}}$ such that derivations with values in these ideals are automatically inner. Derivations with values in ideals of von Neumann algebras has been widely studied in the last decades (see e.g. [6, 10,11,12,13,14, 19, 32, 33, 37, 44, 54, 55], although this list is far from a comprehensive list of references). For completeness, we recall some of the established results for derivations with values in ideals.

For an arbitrary von Neumann algebra ${{\mathcal {M}}}$ and any (not necessarily closed) ideal ${{\mathcal {E}}}$ of ${{\mathcal {M}}}$, it is known [11, 12] that any derivation $\delta :{{\mathcal {M}}}\rightarrow {{\mathcal {E}}}$ is inner. However, when one considers derivations $\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$, where ${{\mathcal {A}}}$ is a von Neumann subalgebra of ${{\mathcal {M}}}$, there are examples of non-inner derivations [55, Theorem 1.2]. For a semifinite von Neumann algebra ${{\mathcal {M}}}$ equipped with a faithful normal semifinite trace $\tau$, and a $C^*$-subalgebra ${{\mathcal {A}}}$ of ${{\mathcal {M}}}$, it is proved in [10] that a derivation $\delta$ defined on ${{\mathcal {A}}}$ is necessarily inner provided that the values $\delta$ belong to the ideal $E({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$ generated by a fully symmetric function space $E(0,\infty )$ having the Fatou property and order continuous norm (see e.g. Sect. 2 and [28] for precise definitions). In Theorem 8.3 below, we show that the same result holds for derivations $\delta :{{\mathcal {A}}}\rightarrow E({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$ with significantly weaker conditions on the symmetric space $E({{\mathcal {M}}},\tau )$ provided that ${{\mathcal {A}}}$ is a von Neumann subalgebra of ${{\mathcal {M}}}$. The conditions we impose on $E({{\mathcal {M}}},\tau )$ is that $E({{\mathcal {M}}},\tau )\subset S_0({{\mathcal {M}}},\tau )$ is a strongly symmetric space having the Fatou property. Thus, we significantly extend [44, Theorem 14].

We note that the Fatou property is an analogue of the so-called “dual normal” property of bimodules over von Neumann algebras (see e.g. [21] and [61]). It is known that every derivation from a hyperfinite von Neumann algebra ${{\mathcal {A}}}$ into a dual normal ${{\mathcal {A}}}$-bimodule is inner (see e.g. [56, Theorem 2] and [61, Theorem 2.4.3]). However, no additional conditions on the von Neumann subalgebra are needed in our setting.

Theorem 8.3

Let ${{\mathcal {M}}}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$, let $E({{\mathcal {M}}},\tau )$ be a strongly symmetric space with the Fatou property and let ${{\mathcal {A}}}$ be a von Neumann subalgebra of ${{\mathcal {M}}}$. Then every derivation $\delta$ from ${{\mathcal {A}}}$ into $E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ is necessarily inner, that is, there exists $T\in E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ such that $\delta =\delta _T$.

Proof

For any symmetric space $E({{\mathcal {M}}},\tau )$ such that $E({{\mathcal {M}}},\tau )\nsubseteq S_0({{\mathcal {M}}},\tau )$, there exists an element $X\in E({{\mathcal {M}}},\tau )$ such that $\mu (X) \ge \alpha \chi _{(0,\infty )}\ge \alpha \mu (\mathbf{1})$ for some $\alpha >0$ (see Sect. 2.3 and Definition 2.1). By Definition 2.5, we obtain that $\mathbf{1}\in E({{\mathcal {M}}},\tau )$, which implies that ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )\subset {{\mathcal {M}}}\subset E({{\mathcal {M}}},\tau )$ (see Sect. 2.2). That is, $E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) ={{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$. By Theorem 8.1, it is sufficient to prove the case when $E({{\mathcal {M}}},\tau )\subset S_0({{\mathcal {M}}},\tau )$. In particular, $E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau ) =E({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$.

We first assume that the carrier projection of $E({{\mathcal {M}}},\tau )$ is $\mathbf{1}$. By [28, Theorem 32], we have that $E({{\mathcal {M}}},\tau )=E({{\mathcal {M}}},\tau )^{\times \times }$, that is, $E({{\mathcal {M}}},\tau )$ is the Köthe dual of $E({{\mathcal {M}}},\tau )^{\times }$. Since $E({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}\subset {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$, it follows from Theorem 8.1 and Proposition 3.10 that there is a $T\in K_\delta$ such that $\delta =\delta _T$. Hence, there exists a net $\{T_i\} \subset co\{U\delta (U^*) \mid U\in {{\mathcal {U}}}({{\mathcal {A}}})\}\subset E({{\mathcal {M}}},\tau )\cap {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ such that $T_i \rightarrow _{so} T$ with

$$\begin{aligned} \sup _i \tau (|T_i X| ) \le \sup _i \left\| T_i\right\| _E \le \left\| \delta \right\| _{{{\mathcal {A}}}\rightarrow E}< \infty \end{aligned}$$

(8.1)

for every X in the unit ball of $E({{\mathcal {M}}},\tau )^\times$ (see Sect. 2.2 and [56, Theorem 2]).

Fix $X\in E({{\mathcal {M}}},\tau )^\times$ with $\left\| X\right\| _{E^\times } \le 1$. Let Z be an arbitrary operator in $L_1({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$ such that $Z\prec \prec X$. Since $E({{\mathcal {M}}},\tau )^\times$ is a fully symmetric space (see e.g. [28, Theorem 27] or [27, Proposition 3.7]), it follows that $Z\in E({{\mathcal {M}}},\tau )^\times$ and $\Vert Z\Vert _{E^\times } \le 1$. Hence, $\Vert T_iZ\Vert _1=\tau (|T_iZ|) {\mathop {\le }\limits ^{(8.1)}} \Vert \delta \Vert _{{{\mathcal {A}}}\rightarrow E}<\infty$. Since $\Vert T_i\Vert _\infty \le \Vert \delta \Vert _{{{\mathcal {A}}}\rightarrow {{\mathcal {M}}}}<\infty$ (see [56, Theorem 2]) and $Z\in L_1({{\mathcal {M}}},\tau )$, it follows from [3, Lemma 2.5] that $TZ \in L_1({{\mathcal {M}}},\tau )$ with $\Vert TZ\Vert _1\le \Vert \delta \Vert _{{{\mathcal {A}}}\rightarrow E} <\infty$. Noting that $X,T \in L_1({{\mathcal {M}}},\tau ) + {{\mathcal {M}}}$ (see e.g. [28, Lemma 25]), it follows from [25, Theorems 3.10 and 4.12] that

$$\begin{aligned} \tau (|TX| ) \le \sup \{\tau (|TZ|): Z \in L_1({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}, Z\prec \prec X\} \le \Vert \delta \Vert _{{{\mathcal {A}}}\rightarrow E}. \end{aligned}$$

Since $X\in E^\times ({{\mathcal {M}}},\tau )$, $\Vert X\Vert _{E^\times } \le 1$, is arbitrary and $E({{\mathcal {M}}},\tau )= (E({{\mathcal {M}}},\tau )^\times )^\times$, it follows that $T\in E({{\mathcal {M}}},\tau )$, as required.

Now, consider the general case. Let $c_E$ be the carrier projection of $E({{\mathcal {M}}},\tau )$. Then, ${{\mathcal {M}}}_{c_E}$ is a von Neumann algebra with identity $c_E$. By Corollary 8.2, there is a $T\in {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ such that $\delta =\delta _T$ on ${{\mathcal {A}}}$. Note that $c_E$ is a central projection in ${{\mathcal {M}}}$ (see [28, Corollary 6]). Hence, $E({{\mathcal {M}}}_{c_E},\tau ):=E({{\mathcal {M}}},\tau ) \subset S({{\mathcal {M}}}_{c_E},\tau )$ is a strongly symmetric space having the Fatou property and $\delta _T: {{\mathcal {A}}}_{c_E} \rightarrow E({{\mathcal {M}}}_{c_E},\tau )\cap {{\mathcal {M}}}_{c_E}$ is also a derivation. By the first part of the proof, there is a $K\in E({{\mathcal {M}}}_{c_E},\tau )\cap {{\mathcal {M}}}_{c_E}$ such that $\delta _T=\delta _K$ on ${{\mathcal {A}}}_{c_E}$. For every $X\in {{\mathcal {A}}}$, we have $\delta _T( X)\in E({{\mathcal {M}}},\tau )$ and therefore $c_E \delta _T( X)=\delta _T( X)$ (see [28, Corollary 6]). Hence, for every $X\in {{\mathcal {A}}}$, we have $\delta _T(c_E^{\perp } X) = c_E^\perp \delta _T( X) = c_E^\perp c_E\delta _T( X)=0$ and therefore,

$$\begin{aligned} \delta (X) =\delta _T(X) = \delta _T((c_E +c_E^{\perp }) X) = \delta _T(c_E X)+\delta _T(c_E^{\perp } X) =\delta _K(c_E X). \end{aligned}$$

Since $c_E$ is a central projection in ${{\mathcal {M}}}$ and $c_E K=K =K c_E$, it follows that $\delta (X) =\delta _K(c_E X)=\delta _K(X)$. Noting that $K\in E({{\mathcal {M}}}_{c_E},\tau )\cap {{\mathcal {M}}}_{c_E} = E({{\mathcal {M}}},\tau )\cap {{\mathcal {M}}}$, we complete the proof. $\square$

Since the non-commutative $L_p$-spaces ($1\le p\le \infty$) are strongly symmetric spaces with the Fatou property (see e.g. [28, Section 3.4]), the above theorem is a unification and extension of the results due to Johnson and Parrott [37], due to Kaftal and Weiss [44] and due to Popa [54]. Actually, the majority of Banach symmetric spaces used in analysis are strongly symmetric, and most of them have the Fatou property. Theorem 8.3 give an affirmative answer to Question 1.1 for ‘almost’ every proper symmetric ideal ${{\mathcal {E}}}$ in ${{\mathcal {M}}}$, that is, for most proper symmetric ideals ${{\mathcal {E}}}$ in ${{\mathcal {M}}}$, derivations from an arbitrary von Neumann subalgebra of ${{\mathcal {M}}}$ into ${{\mathcal {E}}}$ are automatically inner. One should note that the class of symmetric ideals characterized in this paper covers almost every ideal ${{\mathcal {E}}}$ corresponding to a symmetric function space in the sense of Calkin (see [45, 47]).

In the meantime, the ideal ${{\mathcal {J}}}({{\mathcal {M}}})$ of all compact operators in ${{\mathcal {M}}}$ is not corresponding to any symmetric function space whenever ${{\mathcal {M}}}\ne {{\mathcal {J}}}({{\mathcal {M}}})\ne {{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ (see e.g. [55, Section 8]).

Appendix A. Some Remarks On the Proof of [55]

When considering derivations into the ideal ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ of all $\tau$-compact operators in ${{\mathcal {M}}}$, it was natural that the remarkable paper [55] on derivations into the ideal ${{\mathcal {J}}}({{\mathcal {M}}})$ of all compact operators in ${{\mathcal {M}}}$ came into our mind. At first, we thought that following their proof step by step would be a right way to prove the derivation problem for ${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$ (actually, our proof is different from theirs and we only need the main result in [55] and some techniques used in [55, Section 4]). However, comprehending the delicate proof was a tough job to us. We would like to express our gratitude to those who have helped us in understanding the proof. They are Professors Erik Christensen, Ken Dykema, Marius Junge, Sorin Popa and Florin Rădulescu.

One of the difficulties we encountered is a technical inequality in [55, Section 7.4] (see 10th line from the bottom of Page 510), which took us several months to understand (even though, in our final version, we do not need this technique). Thanks to the hint given by Professor Florin Rădulescu, we are able to “fill the gap” in [55, Section 7.4] in our understanding, using our recent paper [10] and involving an additional reduction step. Our proof is of interest in its own right. We provide our “understanding” below for the sake of the reader.

From now on, we adopt the notations used in [55, Section 7.4]. Let ${\mathcal {M}}$ be a $\sigma$-finite semifinite von Neumann algebra with a semifinite faithful normal trace $\varphi$ (and a normal faithful state $\psi$ on center ${{\mathcal {Z}}}({{\mathcal {M}}})$ of ${{\mathcal {M}}}$) and let ${\mathcal {N}}$ be a separable $II_1$ von Neumann subalgebra of ${\mathcal {M}}$, and ${{\mathcal {N}}}'\cap {{\mathcal {M}}}$ contains no finite projections of ${{\mathcal {M}}}$. Moreover, $\delta : {\mathcal {N}}\longrightarrow {\mathcal {J}}({\mathcal {M}})$ (the ideal generated by all finite projections in ${{\mathcal {M}}}$) is a skew-adjoint derivation such that $\delta ({\mathcal {Z}}({\mathcal {N}}))=\{0\}$ (see [55, The 2nd paragraph of Section 7.4]). Hence, for every $p\in {{\mathcal {Z}}}({{\mathcal {N}}})$, we have $\delta (xp)=p\delta (x)p \in p{{\mathcal {J}}}({{\mathcal {M}}})p$ for every $x\in {{\mathcal {N}}}$.

Recall that $K_\delta := {\overline{co}}^{wo}\{\delta (v)v^* : v\in U({{\mathcal {N}}})\}$, ${{\mathcal {M}}}_\varphi :=\{x\in {{\mathcal {M}}}\mid \varphi (x^*x )<\infty \}$ and ${{\mathcal {M}}}_{\varphi ,\psi }^1 = \{x\in {{\mathcal {M}}}_\varphi \mid \Vert x\Vert _\infty \le 1, \varphi (x^*xp)\le \psi (p) \text{ for } \text{ all } p\in {{\mathcal {Z}}}({{\mathcal {M}}})\}$. For a $T\in {{\mathcal {M}}}$, $\delta _T$ is defined by $[T,\cdot ]$.

If one can prove that $\delta =0$, then every derivation from ${{\mathcal {N}}}$ into ${{\mathcal {J}}}({{\mathcal {M}}})$ is inner. Assume by contradiction that $\delta$ is not inner (therefore, $\delta \ne 0$). In [55], it is asserted show that there is a projection $p\in {{\mathcal {Z}}}({{\mathcal {N}}})$ and a projection $e\in {{\mathcal {M}}}_p$ such that $\varphi (e(vp)^*\delta (vp)e)\ne 0$ for some unitary element $v\in {{\mathcal {N}}}$, ${{\mathcal {N}}}_p \ni y \mapsto \varphi (ye)$ is faithful on ${{\mathcal {N}}}_p$ and e is a finite sum of elements in ${{\mathcal {M}}}_{\varphi ,\psi }^1$ (see 10th line from the bottom of [55, Page 510]). However, the authors of [55] did not provide detailed proof for the above assertion. With the help of Professor Florin Rădulescu, we complete the proof for the assertion, which is displayed below.

Before proceeding to the proof, we need the following proposition.

Proposition A.1

There exists a projection $p\in {\mathcal {Z}}({\mathcal {N}})$ such that the derivation $\delta _{(p)} : {\mathcal {N}}p\longrightarrow p{\mathcal {J}}({\mathcal {M}})p$ (defined by $\delta (pxp)$ for every $x\in {{\mathcal {N}}}$) is inner and there is no projection $q\le \mathbf{1}-p$ such that $\delta _{(q)} :{\mathcal {N}}q\longrightarrow q{\mathcal {J}}({\mathcal {M}})q$ is inner.

Proof

If there are no projections $p\in {{\mathcal {Z}}}({{\mathcal {N}}})$ such that the derivation $\delta _{(p)} : {\mathcal {N}}p\longrightarrow p{\mathcal {J}}({\mathcal {M}})p$ is inner, then we set $p=0$.

In the following, we assume that there exists such a non-zero projection.

Note that the subalgebra ${\mathcal {N}}$ is separable. The Zorn lemma implies that there exists a maximal family of non-zero pairwise disjoint projections $\{q_i\}_{i=1}^N\subset {\mathcal {Z}}({\mathcal {N}})$, where $N=\infty$, or N is a positive integer, such that

$$\begin{aligned} \delta _{(q_i)}(x)=[d_i,x],\ x\in {\mathcal {N}}_{q_i} \end{aligned}$$

for some $d_i\in q_i{\mathcal {J}}({\mathcal {M}})q_i$ (note that we can say $x\in {{\mathcal {N}}}$ here). Moreover, [55, Lemma 4.6] implies that there exists $d_i'\in q_i K_\delta$ such that $\delta _{d_i}=\delta _{d_i'}$. Then, $\Vert d_i'\Vert \le \Vert \delta \Vert$. For the sake of convenience, we still denote $d_i'$ by $d_i$. Let

$$\begin{aligned} p_i:=\vee _{j \le i} q_j, \text{ and } p=wo-\lim _i p_i. \end{aligned}$$

Note that $D_i:= \sum _{j\le i } d_j \in K_\delta$ (see Remark 3.7) and $p_i \delta =\delta _{D_i}$. Since the supports of $d_i$ are pairwise disjoint and every $D_i$ is bounded with $\Vert D_i\Vert _\infty \le \Vert \delta \Vert _\infty$, it follows that $D_i \rightarrow _{wo} D$ for some $D\in {{\mathcal {M}}}$. Moreover, since every $D_i$ is in $K_\delta$, it follows from the definition of $K_\delta$ that $D\in K_\delta$. Note that $p D_i = D_i p=D_i$. That is, $D_i = p D_i \rightarrow _{wo}pD$. Hence, for every $x\in {{\mathcal {N}}}$, we have

$$\begin{aligned} p \delta (x) =wo-\lim _i p_i\delta (x) =wo-\lim _i \delta _{D_i}(x) =\delta _{pD}(x) . \end{aligned}$$

Moreover, by [55, Proposition 4.4], we have $pD\in p {{\mathcal {J}}}({{\mathcal {M}}}) p$. That is, $\delta _{(p)}$ is inner.

The maximality of the family $\{q_i\}_{i\in I}$ implies that there is no projection $q\le \mathbf{1}-p$ such that $\delta _{(q)}:{\mathcal {N}}q\longrightarrow q{\mathcal {J}}({\mathcal {M}})q$ is inner. $\square$

Now, back to [55, Section 7.4]. Let p be the projection constructed in Proposition A.1. Since we have assumed that $\delta$ is not inner, it follows that we may suppose now that ${\mathbf {1}}-p>0$ and consider the reduced algebra $({\mathbf {1}}-p){\mathcal {M}}({\mathbf {1}}-p)$.

All conditions of [55, Section 7.4] are preserved in this reduced algebra. Indeed, let nonzero projection ${\mathcal {M}}\ni q\le {\mathbf {1}}-p$ and $q\in ({\mathcal {N}}({\mathbf {1}}-p))'$. Then, $q\in {\mathcal {N}}'\cap {\mathcal {M}}$ and therefore is an infinite projection (since ${{\mathcal {N}}}'\cap {{\mathcal {M}}}$ contains no finite projections of ${{\mathcal {M}}}$). In addition, the derivation $\delta$ vanishes on a maximal abelian subalgebra ${\mathcal {A}}({\mathbf {1}}-p)\subset {{\mathcal {N}}}({\mathbf {1}}-p)$.

Take a $\varphi$-finite projection $e\in (\mathbf{1}-p){{\mathcal {M}}}_{\varphi ,\psi }^1(\mathbf{1}-p)$. Now, we can construct a projection $p_n\in (\mathbf{1}-p){{\mathcal {Z}}}({{\mathcal {N}}})(\mathbf{1}-p)$ and a projection $e_n\in p_n{{\mathcal {M}}}p_n$ (which is a finite sum of elements in ${{\mathcal {M}}}_{\varphi ,\psi }^1$) such that the form $x \mapsto \varphi (xe_n)$ is faithful on ${\mathcal {N}}p_n$. Indeed, let $q\in {{\mathcal {N}}}_{\mathbf{1}-p}$ be the support of the normal form ${{\mathcal {N}}}_{\mathbf{1}-p}\ni y\mapsto \varphi (ye)$ as in [55, Section 7.4]. By Proposition A.2 below, there is a central projection $p_n\in {{\mathcal {Z}}}({{\mathcal {N}}}_{\mathbf{1}-p})$ and a finite number of unitary elements $u_1,\cdots , u_{k(n)}$ in ${{\mathcal {N}}}_{\mathbf{1}-p}$ such that

$$\begin{aligned} \vee _i u_i p_n q u_i^* = p_n. \end{aligned}$$

(A.1)

Let $e_n$ be the support projection of $\sum _i u_i p_n e p_n u_i^*$. Since $e\in (\mathbf{1}-p){{\mathcal {M}}}_{\varphi ,\psi }^1(\mathbf{1}-p)$, it follows that $e_n$ is a finite sum of elements in ${{\mathcal {M}}}_{\varphi ,\psi }^1$ (see e.g. [55, 2.2.2]). Now, we prove the faithfulness of ${{\mathcal {N}}}p_n\ni y \mapsto \varphi (y e_n)$. Let $0\le y\in {{\mathcal {N}}}p_n$ and $\varphi (ye_n)=0$. Therefore,

$$\begin{aligned} \varphi \left( y\sum _i u_i p_n e p_n u^*_i\right)&= \varphi \left( y^{1/2}(\sum _i u_i p_n e p_n u^*_i ) y^{1/2}\right) \\&\le \varphi \left( y^{1/2} \left\| \sum _i u_i p_n e p_n u^*_i \right\| _\infty e_n y^{1/2}\right) =0. \end{aligned}$$

Since $\varphi (yu_i p_n ep_n u^*_i)\ge 0$ for all i, we have that $\varphi (yu_i p_n ep_n u^*_i)=0$ for all i. Therefore, we obtain that $yu_ie=yu_i p_n e=0$, which implies that $u_i^*yu_ie=0$. Hence, we have $s(u_i^*yu_i)e=0$ and therefore, $\varphi (s(u_i^*yu_i)e)=0$. Then, $s(u_i^*yu_i)\le {\mathbf {1}}-q$ and $u_i^*yu_iq=0$. Hence, we have $yu_iqp_nu_i^*=0$, which implies that $y=yp_n=y\bigvee _i u_iqp_nu_i^*=0$.

Since $\delta$ is a skew-adjoint derivation, it follows that $p_n v^*\delta (v)p_n$ is self-adjoint for every $v\in p_n {{\mathcal {U}}}({{\mathcal {N}}}) p_n$.

There are two possible cases:

Case 1.:

There exists $v\in {\mathcal {U}}({\mathcal {N}})$ such that the support $s(p_nv^*\delta (v)p_n)$ of $p_nv^*\delta (v)p_n$ is a $\varphi$-infinite projection.

Case 2.:

For every $v\in {\mathcal {U}}({\mathcal {N}})$, $s(p_nv^*\delta (v)p_n)$ is a $\varphi$-finite projection.

If $\varphi (e_n p_nv^*\delta (v)p_n )\ne 0$ for some unitary element v in ${{\mathcal {N}}}$, then this v is exactly the unitary element that we need. That is, $p_n$, v and $e_n$ satisfy the assumptions in [55, Section 7.4], which implies that $\delta _{(p_n)}$ is inner. This contradicts the choice of p. Hence, we may always assume that $\varphi (e_n p_nv^*\delta (v)p_n ) = 0$ for all $v\in {{\mathcal {U}}}({{\mathcal {N}}})$.

Case 1.

Suppose that $v\in {\mathcal {U}}({\mathcal {N}})$ is such that $s(p_nv^*\delta (v)p_n)$ is a $\varphi$-infinite projection. Set

$$\begin{aligned} x:=p_nv^*\delta (v)p_n. \end{aligned}$$

That is, $\varphi (e_nx)=0$. Since $\delta$ is skew-adjoint, it follows that $x^*=x$.

• (a) Suppose that $f:=(p_n-e_n)\wedge s(x_+)\ne 0$. Then, we have

  $$\begin{aligned} \varphi (fx)=\varphi (fx_+)>0. \end{aligned}$$

  By [55, 2.2.4], there exists a $\varphi$-finite projection $g\in {{\mathcal {M}}}_{\varphi ,\psi }^1$ with $g\le f$ such that $\varphi (gx)=\varphi (gx_+)\ne 0$. In particular, $g\le f\le s(x_+)$. We have that

  $$\begin{aligned} \varphi ((e_n+g)x)=\varphi (e_nx)+\varphi (gx)=0+ \varphi (gx)\ne 0. \end{aligned}$$

  Since $\varphi (\cdot e_n)$ is faithful on ${{\mathcal {N}}}_{p_n}$, it follows that the form $y \mapsto \varphi (y(e_n+g))$ is faithful on ${{\mathcal {N}}}_{p_n}$. Hence, v, $e_n+g$ and $p_n$ are what we need. Similarly, we can consider the case when $(p_n-e_n)\wedge s(x_-)\ne 0$.

• (b). The remaining case is $(p_n -e_n)\wedge s(x_+) = 0 = (p_n -e_n)\wedge s(x_-)$. In this case, since

  $$\begin{aligned} e_n\ge (p_n-e_n)\vee s(x_+) - (p_n-e_n) \sim s(x_+)- (p_n -e_n)\wedge s(x_+) =s(x_+), \end{aligned}$$

  it follows that $s(x_+)\preceq e_n$ (and similarly, $s(x_-)\preceq e_n$). Since $e_n$ is $\varphi$-finite, it follows that $s(x_+)$ and $s(x_-)$ are $\varphi$-finite projections. Therefore, $s(x)=s(x_+)+s(x_-)$ is a $\varphi$-finite projection, which contradicts the choice of v.

Case 2.

This is the case, where the result in our recent paper [10] plays a crucial role.

Assume that for any $v\in {\mathcal {U}}({\mathcal {N}})$, the projection $s(p_nv^*\delta (v)p_n)$ is $\varphi$-finite. We have that

$$\begin{aligned} p_nv^*\delta (v)p_n=p_nv^*\delta (v) p_ns(p_nv^*\delta (v)p_n)\in p_nL_2({\mathcal {M}},\varphi )p_n\cap p_n{{\mathcal {M}}}p_n \end{aligned}$$

and

$$\begin{aligned} p_n\delta (v)p_n=p_nvp_n\cdot p_n v^*\delta (v)p_n\in p_nL_2({\mathcal {M}},\varphi )p_n\cap p_n{{\mathcal {M}}}p_n \end{aligned}$$

for any $v\in {\mathcal {U}}({\mathcal {N}})$. Therefore, the image $\delta$ lies in $p_nL_2({\mathcal {M}},\varphi )p_n\cap p_n{{\mathcal {M}}}p_n$.

Now, [10, Corollary 3.6] implies that there exists $d\in p_n L_2({\mathcal {M}},\varphi )p_n\cap p_n{{\mathcal {M}}}p_n\subset p_n{{\mathcal {J}}}({{\mathcal {M}}})p_n$ such that $\delta =[d,\cdot ]$ on ${{\mathcal {N}}}_{p_n}$. Since $p_n \le \mathbf{1}-p$, we obtain a contradiction to the maximality of p.

Thus, supposing that ${\mathbf {1}}-p>0$, we obtain either contradictions or that there is a projection $p\in {{\mathcal {Z}}}({{\mathcal {N}}})$ and a projection $e\in {{\mathcal {M}}}_p$ such that $\varphi (e(vp)^*\delta (vp)e)\ne 0$ for some unitary element $v\in {{\mathcal {N}}}$, ${{\mathcal {N}}}_p \ni y \mapsto \varphi (ye)$ is faithful on ${{\mathcal {N}}}_p$ and e is a finite sum of elements in ${{\mathcal {M}}}_{\varphi ,\psi }^1$.

For the sake of completeness, we present a proof for (A.1).

Proposition A.2

Let ${{\mathcal {N}}}$ be a finite von Neumann algebra and let q be a projection in ${{\mathcal {N}}}$. There exists a sequence of central projections $p_n$ in ${{\mathcal {N}}}$, such that $p_n\uparrow c(q)$ (c(q) is the central support of q), and

$$\begin{aligned} p_n=\bigvee _{i=1}^{n+1}u_{n,i}^*p_nqu_{n,i}, \end{aligned}$$

where $u_{n,1},\dots ,u_{n,n+1}$ are unitaries from ${{\mathcal {N}}}$ for every $n\ge 1$.

Proof

Passing, if necessary, to the algebra ${{\mathcal {N}}}_{c(q)}$, we can assume that $c(q)={\mathbf {1}}$.

The main tool in the proof is the theory of comparison of projections.

We firstly construct a sequence of pairwise disjoint central projections $\{z_n\}$ in ${{\mathcal {N}}}$ and pairwise disjoint projections $\{r_n\}$ inductively.

• 1. By [42, Theorem 6.2.7], there exists z such that $zq\succeq z({\mathbf {1}}-q)$ and $({\mathbf {1}}-z)q\preceq ({\mathbf {1}}-z)({\mathbf {1}}-q)$. We define $z_1:=z$ and define $r_1$ to be the projection such that $({\mathbf {1}}-z_1)({\mathbf {1}}-q)\ge r_1\sim ({\mathbf {1}}-z_1)q$.

• 2. Assume now that $z_1,\dots ,z_n$ and $r_1,\dots ,r_n$ are already defined. Consider projections $({\mathbf {1}}-z_1-\dots -z_n) q$ and $({\mathbf {1}}-z_1-\dots -z_n) ({\mathbf {1}}-q-r_1-\dots -r_n)$. [42, Theorem 6.2.7] applied in the reduced von Neumann algebra ${{\mathcal {N}}}_{{\mathbf {1}}-z_1-\dots -z_n}$ guarantees that there exist $z_{n+1}\le {\mathbf {1}}-z_1-\dots -z_n$ and $r_{n+1}\le ({\mathbf {1}}-z_1-\dots -z_{n+1})({\mathbf {1}}-q-r_1-\dots -r_n)$, such that $z_{n+1}q\succeq z_{n+1}({\mathbf {1}}-q-r_1-\dots -r_n)$ and $({\mathbf {1}}-z_1-\dots -z_{n+1})q\sim r_{n+1}$. By construction $\{r_{n}\}$ are pairwise disjoint.

We claim that $\sum _{n=1}^\infty z_n=\mathbf{1}$. Suppose that $z_0:={\mathbf {1}}-\sum _{n=1}^\infty z_n\ne 0$. Since $c(q)={\mathbf {1}}$, it follows that $qz_0\ne 0$. By the construction, $z_0 r_m \sim z_0 q\preceq z_0({\mathbf {1}}-q-r_1-\dots -r_n)$ for all $n,m\ge 1$. Let $\tau _c$ be the faithful centervalued trace on ${{\mathcal {N}}}$. We have that

$$\begin{aligned} \tau _c(z_0q)\le \tau _c(z_0({\mathbf {1}}-q-r_1-\dots -r_n)) = \tau _c(z_0) - (n+1)\tau _c(z_0q), \end{aligned}$$

that is, $0\le (n+2)\tau _c(qz_0)\le \tau _c(z_0)$, and therefore $(n+2)\Vert \tau _c(qz_0)\Vert \le 1$, for all $n\ge 1$. This contradiction implies that $\sum _{n=1}^\infty z_n={\mathbf {1}}$.

We set $p_n=\sum _{i=1}^n z_n$. It is clear that $p_n\uparrow {\mathbf {1}}$.

Note that $z_n=z_nq+z_n({\mathbf {1}}-q-r_1-\dots -r_{n-1})+z_nr_1+\dots +z_nr_{n-1}$. All summands on the right side are pairwise orthogonal and majorized by $z_nq$. Hence, for every $n\ge 1$ and $k\le n$ the projection $z_k$ can be decomposed in the sum of no more than $n+1$ pairwise disjoint summands majorized by $z_kq$. Therefore, $p_n=\sum _{i=1}^{n+1}e_i$, where $e_i$ are pairwise orthogonal projections and $e_i \preceq p_nq,\ i=1,\dots ,n+1$.

Since ${{\mathcal {N}}}$ is a finite von Neumann algebra, there exist unitaries $u_{n,1},\dots ,u_{n,{n+1}}$ from ${{\mathcal {N}}}$, such that $e_i\le u_{n,i}^*p_nqu_{n,i},\ i=1,\dots ,n+1$ (see e.g. [64, Chapter XIV, Lemma 2.1]). Hence, we obtain that

$$\begin{aligned} p_n=\bigvee _{i=1}^{n+1}u_{n,i}^*p_nqu_{n,i}. \end{aligned}$$

$\square$