Real ideals of compact operators of complex factors

Abstract

Real ideals of compact operators for (complex) factors are investigated. A description (up to isomorphisms) of real two-sided ideals of relatively compact operators of a complex W*-factors is given. A relative weak (RW)$_r$ convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is extended to the compact operators in semifinite real W*-algebras.

1 Introduction

In the present paper we investigate the real ideals of compact operators for complex factors and give a description (up to isomorphisms) of real two-sided ideal of relatively compact operators of the complex W*-factors. A concept of relative weak (RW)$_r$ convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is extended to the compact operators in semifinite real W*-algebras.

2 Preliminaries

Let $B(H)$ be the algebra of all bounded linear operators on a complex separable Hilbert space $H$. A weakly closed *-subalgebra $\mathfrak A$ containing the identity operator $\mathbf{1}\!\!\mathrm{I}$ in $B(H)$ is called a W$^*$-algebra. A real *-subalgebra $R\subset B(H)$ is called a real W$^*$-algebra if it is closed in the weak operator topology, $\mathbf{1}\!\!\mathrm{I}\in R$ and $R\cap iR=\{0\}$. A real W$^*$-algebra $R$ is called a real factor if its center $Z(R)$ consists of the elements $\{\lambda \mathbf{1}\!\!\mathrm{I}, \lambda \in \mathbb{R }\}$, where $\mathbb{R }$ is the field of all real numbers. We say that a real W$^*$-algebra $R$ is of the type I$_{fin}$, I$_{\infty }$, II$_1,$ II$_{\infty },$ or III$_{\lambda }$, $(0 \le \lambda \le 1)$ if the enveloping W$^*$-algebra $\mathfrak{A }(R)$ has the corresponding type in the ordinary classification of W$^*$-algebras. A linear mapping $\alpha$ of an algebra into itself with $\alpha (x^*)=\alpha (x)^*$ is called an *-automorphism if $\alpha (xy)=\alpha (x)\alpha (y)$; it is called an involutive *-antiautomorphism if $\alpha (xy)=\alpha (y)\alpha (x)$ and $\alpha ^2(x)=x$. If $\alpha$ is an involutive *-antiautomorphism of a W$^*$-algebra $M$, we denote by $(M,\alpha )$ the real W$^*$-algebra generated by $\alpha$, i.e. $(M,\alpha )=\{x\in M: \ \alpha (x)=x^*\}$. Conversely, every real W*-algebra $R$ is of the form $(M,\alpha )$, where $M$ is the complex envelope of $R$ and $\alpha$ is an involutive *-antiautomorphism of $M$ (see [1, 2, 5, 9]). Therefore we shall identify from now on the real von Neumann algebra $R$ with the pair $(M,\alpha )$.

A trace on a (complex or real) W*-algebra $N$ is a linear function $\tau$ on the set $N^+$ of positive elements of $N$ with values in $[0,+\infty ]$, satisfying $\tau (uxu^*)=\tau (x)$, for an arbitrary unitary $u$ and for any $x$ in $N$.

The trace $\tau$ is said to be finite, if $\tau (\mathbf{1}\!\!\mathrm{I})<+\infty$; semifinite, if given any $x\in N^+$ there is a nonzero $y\in N^+$, $y\le x$ with $\tau (y)<+\infty$.

3 Real and complex ideals of W*-algebras

Definition1

Let $M$ be a W*-algebra. A real subspace $I$ of $M$ is called a real ideal of $M$ if $I\cdot M \subset I_c$, where $I_c$ is the smallest complex subspace of $M$, containing $I$.

It is easy to see that the subspace $I_c$ is equal to $I+iI$, therefore a real subspace $I$ is a real ideal if and only if $I\cdot M \subset I+iI$.

Since each complex subspace of $M$ is a real subspace, any complex ideal is automatically a real ideal of $M$. Let $I$ be a real ideal of $M$. If there exists a real W*-subalgebra $R$ of $M$ with $R+iR=M$, such that $I\subset R$, then $I$ is called a pure real ideal of $M$. In this case, it is obvious that we have $I\cdot R\subset I$. Note that, the reverse is not true, i.e. from $I\cdot R\subset I$ it does not follow $I\subset R$. But a complex subspace $J=I+iI$ always is a complex ideal of $M$. On the other hand if $I\subset R$ is a real subspace of $M$ and $I+iI$ is a complex ideal, then $I$ is a pure real ideal, i.e. we obtain $I\cdot R\subset I$.

Let, now $I$ and $Q$ be pure real ideals of $M$. In general, the set $I+iQ$ is not a (complex) subspace. More precisely the set $I+iQ$ is a complex subspace if and only if $I=Q$. Therefore we consider the smallest complex subspace $J$ of $M$, containing $I$ and $Q$. Obviously $J$ is equal to $(I+Q)+i(I+Q)$. Thus, if $I$ and $Q$ are real ideals, then $J=(I+Q)+i(I+Q)$ is a complex ideal.

4 Ideals of compact operators

Let $(M,\alpha )$ be a real factor and let $\tau$ be an $\alpha$-invariant semifinite trace on $M$. A subspace $K\subset H$ is called $\tau$-finite (or finite relative to $\tau$), if $\tau (P_K)<+\infty$, where $P_K$ is the canonical projection of $H$ on $K$ with $P_K\in M$.

Now, let $K$ be a subset of $H$. A subset $K$ is called $\tau$-compact (or compact relative to $\tau$), if K is approximated in the norm $\Vert \cdot \Vert _H$ by a bounded sequence of $\tau$-finite subspaces.

A real operator $A$ on $H$ (i.e. $A\in (M,\alpha )$) is called real compact relative to $\tau$ if it is an operator mapping bounded sets into relatively compact sets. We denote by $I$ (respectively, by $J$) the set of all relatively compact operators of $(M,\alpha )$ (respectively, of $M$). Let us recall the following result.

Theorem1

([3, 10, 11]) Let $M$ be a semifinite factor and let $\alpha$ be an involutive *-antiautomorphism of $M$. Then $I$ (respectively, $J$) is a unique (nonzero) uniformly closed two-sided ideal of $(M,\alpha )$ (respectively, of $M$), and $I+iI=J$.

Now, let us recall [4] the notion of the crossed product of a W*-algebra $M$ by a locally compact topological group $G$ by its *-automorphism. Let $\gamma : G\rightarrow Aut(M)$ be a group homomorphism such that the map $g\rightarrow \gamma _g$ is strongly continuous. Let $L_2(G,H)$ be the Hilbert space of all $H$-valued square integrable functions on $G$. We consider a *-algebra $N\subset B(L_2(G,H))$, generated by operators of the form: $\pi _\gamma (a)$ and $u(g)$, where $a\in M$, $g\in G$, and

$$\begin{aligned} \left(\pi _\gamma (a)\xi \right) = \gamma _h^{-1}(a)\xi (h), \quad \left(u(g)\xi \right)(h) = \xi (g^{-1}h), \end{aligned}$$

$\xi =\xi (h)\in L_2(G,H), \ g,h\in G$. The algebra $N$ is called the crossed product of $M$ by $G$, and denoted by $W^*(M,G)$ or $M\times _\gamma G$. Moreover, there exists a canonical embedding $\pi \gamma : M \rightarrow \pi _\gamma (M)\subset N$. Each element $x\in N$ has the form: $x=\sum _{g\in G} \pi _\gamma (x(g))u(g)$, where $x(\cdot )$ is an $M$-valued function on $G$. If $\theta$ is a *-automorphism of $M$, then for the action $\{\theta ^n\}$ of the group $\mathbb{Z }$ on $M$ we denote by $W^*(\theta ,M)$ or $M\times _\theta \mathbb{Z }$ the crossed product of $M$ by $\theta$. Similarly one can define the notion of crossed product for real W*-algebras (see [1, 12–14]).

Let $M$ be a factor of type III$_\lambda$ ($\lambda \not =1$) and let $\alpha$ be an involutive *-antiautomorphism of $M$. Then by [12] (see also [1]), either

• there exist a factor $N$ of type II$_\infty$ and an $\alpha$-invariant automorphism $\theta$ of $N$ such that $(M,\alpha )$ is isomorphic to the (real) crossed product $(N,\alpha )\times _\theta \mathbb{Z }$ or

• there exist a factor $N$ of type II$_\infty$ and an antiautomorphism $\sigma$ of $N$ such that $(M,\alpha )$ is isomorphic to $\bigl ((N\oplus N^{op})\times _\sigma \mathbb{Z }, \beta \bigr )$, where $N^{op}$ is the opposite W*-algebra for $N$ and $\beta (x,y)=(y,x)$, for all $x,y\in N$.

Let’s consider the first case. Let $(M,\alpha )$ be isomorphic to the (real) crossed product $(N,\alpha )\times _\theta \mathbb{Z }$. It is known that $(N,\alpha )\times _\theta \mathbb{Z } + i\cdot (N,\alpha )\times _\theta \mathbb{Z } = N\times _\theta \mathbb{Z }$ (see [12–14]). Let $E:M\rightarrow N$ be the unique $\alpha$-invariant faithful normal conditional expectation (see [15, 16]). Let us state an auxiliary lemma whose proof immediately follows from the linearity and $\alpha$-invariance of $E$.

Lemma1

If $S$ is an ideal in $(M,\alpha )$ and $S_c=S+iS$, then

$$\begin{aligned} E^{-1}(S) + iE^{-1}(S) = E^{-1}(S_c), \quad E^{-1}(S)^+ \subset E^{-1}(S)^+ , \end{aligned}$$

where $E^{-1}(A)=\{x: E(x)\in A\}$.

Recall that a cone $K\subset A^+$ is called hereditary if $x\in A^+$, $y\in K$ and $x\le y$ implies $x\in K$; a subalgebra $B\subset A$ is called hereditary if the cone $B^+$ is hereditary. It is easy to see that any two-sided ideal is hereditary.

Lemma2

If the cone $E^{-1}(S_c)^+$ is hereditary, then the cone $E^{-1}(S)^+$ is also hereditary.

Proof

If $x\!\in \! (M,\alpha )^+\!\subset \! M^+$, $y\!\in \! E^{-1}(S)^+\!\subset \! E^{-1}(S_c)^+$ and $x\!\le \! y$, then $x\!\in \! E^{-1}(S_c)^+$, since $E^{-1}(S_c)^+$ is hereditary. Therefore $\alpha (x)\!=\!x^*$ implies $x\!\in \! E^{-1}(S)^+$. $\square$

Propsition1

Let $M$ be a semifinite factor. If $S\subset (M,\alpha )$ is a two-sided ideal, then the linear span of $E^{-1}(S)^+$ denoted as $span(E^{-1}(S)^+)$ is a hereditary *-subalgebra of $(M,\alpha )$ and a two-sided module over $(M,\alpha )$. Moreover, if $S$ is a norm-closed, then $span(E^{-1}(S)^+)$ is also norm-closed.

Proof

By Lemma 1 and Proposition 3.3 [6] the cone $E^{-1}(S_c)^+$ is hereditary and $span(E^{-1}(S_c)^+)$ is a hereditary *-subalgebra of $M$, where $S_c=S+iS$. By Lemma 2 the cone $E^{-1}(S)^+$ is also hereditary. Using the hereditarity of $span(E^{-1}(S_c)^+)$ one can easily check the hereditarity of $span(E^{-1}(S)^+)$.

If $S$ is norm-closed, then $S^+$ is also norm-closed. The continuity of $E$ implies that $E^{-1}(S)^+$ is closed. Therefore $span(E^{-1}(S)^+)$ is also closed.

Let $x\in E^{-1}(S)^+$ and $y\in (M,\alpha )$. From $x\in E^{-1}(S_c)^+$ and $y\in M$, by Proposition 3.3 [6] we obtain that $yx\in span(E^{-1}(S_c)^+)$, since $span(E^{-1}(S_c)^+)$ is a two-sided module over $M$. On the other hand $yx\in (M,\alpha )$ and $\{a\in span(E^{-1}(S_c)^+) : \ \alpha (a)=a^*\}=span(E^{-1}(S)^+)$. Hence $yx\in span(E^{-1}(S)^+)$. Since the element $x$ is arbitrary from $E^{-1}(S)^+$ by linearity we obtain $yx\in span(E^{-1}(S)^+)$ for any $x\in span(E^{-1}(S)^+)$. Therefore $span(E^{-1}(S)^+)$ is a left-sided module over $(M,\alpha )$. Similarly one can show, that it is a right-sided $(M,\alpha )$-module. $\square$

Now, we put $\mathcal{I}=span\{x\in (M,\alpha ): \ E(x)\in I\}$, where $I$ is the unique (nonzero) uniformly closed two-sided ideal of the semifinite real factor $(N,\alpha )$ (see Theorem 1). By Proposition 1, $\mathcal{I}$ is hereditary.

Lemma 3

The following is valid

$$\begin{aligned} \qquad \quad \mathcal{I }^+ = \{x\in (M,\alpha )^+ : x\le y, \quad {for\, some}\, y\in I^+\}. \end{aligned}$$

Proof

Let $x\in \mathcal{I}^+$ and $J=I+iI$, $\mathcal{J}^+ = \{x\in M^+: x\le z$, for some $z\in J^+\}$. Since $x\in \mathcal{J}^+$ by Proposition 3.7 d) [6], $x\le z$, for some $z\in J^+$. Let $z=y+it$, $y,t\in (N,\alpha )$. Then $y\ge 0$ (because $z\ge 0$) and $z-x=(y-x)+it\ge 0$, hence $y-x\ge 0$, i.e., $x\le y$. Since $I^+\subset J^+$, $y\in I^+$.

Conversely, if $x\in (M,\alpha )^+$ and $x\le y$, for some $y\in I^+$, then again by Proposition 3.7 d) [6] we have $x\in \mathcal{J}^+$. From $\alpha (x)=x^*$ we have $x\in \mathcal{I}^+$. $\square$

From Lemma 3, in particular, it follows, that a projection from $(M,\alpha )$ is finite if and only if it majorized by some finite projection of $(N,\alpha )$.

Let I$_1$ be the norm closure of $\mathcal{I}$. If we apply Proposition 1, Lemma 3 and the scheme of the proofs of Propositions 4.1, 4.5 and Theorem 4.3 [6], then we can prove the following real analogue of Halpern-Kaftal’s theorem.

Theorem2

Let $M$ be a factor of type III$_\lambda$ ($\lambda \not =1$) and let $\alpha$ be an involutive *-antiautomorphism of $M$. If the real factor $(M,\alpha )$ is isomorphic to the (real) crossed product $(N,\alpha )\times _\theta \mathbb Z$, then I$_1$ is a unique (up to an inner automorphism) smallest hereditary real C*-subalgebra of $(M,\alpha )$, containing the ideal $I$, and it is a two-sided module over $(N,\alpha )$.

Let us consider the second case. Let $(M,\alpha )$ be isomorphic to $\left((N\oplus N^{op})\times _\sigma \mathbb{Z }, \beta \right)$, where $N$ is a II$_\infty$-factor, $N^{op}$ is the opposite W*-algebra for $N$ and $\beta (x,y)=(y,x)$, for all $x,y\in N$.

Recall that, a factor $N$ is generated by the fixed point algebra of the one parameter group $\{\sigma ^{\psi }_t: t\in \mathbb{R }\}$ of modular automorphisms, associated with some $\alpha$-invariant faithful normal semifinite weight $\psi$. More precisely, the W*-subalgebra $M_\psi = \{x\in M: \sigma ^\psi _t(x)=x, t\in \mathbb{R }\}$ contains a central projection $p$ such that $N$ is isomorphic to factor $pM_\psi$. In this case the real W*-algebra $(M_\psi , \alpha )$ is isomorphic to $(N\oplus N^{op}, \beta )$ (for more details see [1, 12–14]).

Let $E:(M,\alpha )\rightarrow (N\oplus N^{op}, \beta )$ be a faithful normal conditional expectation (see [15, 16]) and let $J$ be the unique (nonzero) uniformly closed two-sided ideal of the semifinite (complex) factor $N$ (see Theorem 1). Similarly to the first case, we denote by I$_2$ the norm closure of $span\{x\in (M,\alpha ): \ E(x)\in (J\oplus J^{op}, \beta )\}$.

Applying the same reasonings, as in the first case, and the scheme of proofs of Propositions 4.1, 4.5 and Theorem 4.3 [6], we obtain one more real analogue of Halpern-Kaftal’s theorem.

Theorem3

Let $M$ be a factor of type III$_\lambda$ ($\lambda \not =1$) and let $\alpha$ be an involutive *-antiautomorphism of $M$. If the real factor $(M,\alpha )$ is isomorphic to $\left((N\oplus N^{op})\times _\sigma \mathbb{Z }, \beta \right)$, then I$_2$ is the unique (up to inner automorphism) smallest hereditary real C*-subalgebra of $(M,\alpha )$, containing the ideal $I$ and is a two-sided module over $(N,\alpha )$. Here $I$ is the unique (nonzero) uniformly closed two-sided ideal of semifinite real factor $(N,\alpha )$.

Thus, summarizing all above, in the injective case, we can describe all (nonzero) uniformly closed two-sided real ideals of semifinite and pure infinite complex factors. Recall that [1, 5], if $M$ is an injective factor of type II, then there exists a unique conjugacy class of involutive *-antiautomorphisms in $M$; therefore there exists a unique (up to isomorphisms) real subfactor of $M$, generating $M$. If $M$ is an injective factor of type III$_\lambda$ ($0<\lambda <1$), then in $M$ there exist exactly two conjugacy classes of involutive *-antiautomorphism; therefore there exist two (up to isomorphisms) real subfactors of $M$, generating $M$. Hence we obtain the following result.

Theorem4

Let $M$ be a factor. Then the following assertions are true:

• 1. If $M$ is an injective factor of type II$_1$ or type II$_\infty$, then there exist (up to isomorphisms) two (nonzero) uniformly closed two-sided real ideals in $M$. One of them is the complex ideal $J$, the other is the pure real ideal $I$.

• 2. If $M$ is an injective factor of type III$_\lambda$ ($0<\lambda <1$), then there exist (up to isomorphisms) three (nonzero) uniformly closed two-sided real ideals in $M$. One of them is the complex ideal $J$, the two others are the pure real ideals I$_1$ and I$_2$.

5 Relative weak convergence in semifinite real W*-algebras

In this section, we study the relative weak (RW) convergence in a real Hilbert space. We first recall that the elements of the two-sided closed ideal $I$ generated by the projections which are finite relative to a real W*-algebra $(M,\alpha )$ are called compact operators of $(M,\alpha )$.

Let $H_r$ be a real Hilbert space with $H_r+iH_r=H$. A sequence $\{\xi _n\}\subset H_r$ (or $\subset H$) is called weakly converging to $\xi$, if for every projection $P$ which is finite relative to $B(H_r)$ (respectively, $B(H)$), $P\xi _n$ converges strongly to $P\xi$ ($P\xi _n \stackrel{S}{\longrightarrow } P\xi$). This suggests the following generalization:

Definiton2

Let $(M,\alpha )$ be a real W*-algebra in $B(H_r)\subset B(H)=B(H_r)+iB(H_r)$. We say that a sequence $\{\xi _n\}\in H_r$ converges to $\xi$ weakly relative to $(M,\alpha )$ and briefly say (RW)$_r$ converges or $\xi _n \stackrel{(RW)_r}{\longrightarrow } \xi$, if

• 1. $\Vert \xi _n\Vert$ is bounded;

• 2. for every projection $P\in (M,\alpha )$ which is finite relative to $(M,\alpha )$, the sequence $\{P\xi _n\}$ converges strongly to the element $P\xi$, i.e. $P\xi _n \stackrel{S}{\longrightarrow } P\xi$.

Note that a weakly convergent sequence is necessarily bounded, but following the Example 2 of [7], it is easy to construct an example, in which a unbounded sequence satisfies the second condition of Definition 1.

Example

Let $H$ and $K$ be infinite-dimensional separable real Hilbert spaces with orthonormal bases $\{\eta _n\}$, $\{\gamma _n\}$ respectively. We put $R=B(H)\otimes \mathbb{R }\mathbf{1}\!\!\mathrm{I}_K$ and $\xi _n=\sum _{i=n}^{2n}\eta _i\otimes \gamma _i$. Since $\Vert \xi _n\Vert ^2=\sum _{i=n}^{2n}\Vert \eta _i\Vert ^2\Vert \gamma _i\Vert ^2=n$, it is an unbounded sequence. Let $P$ be a finite protection of $R$. Then there is a finite projection $P_0$ in $B(H)$ such that $P=P_0\otimes \mathbf{1}\!\!\mathrm{I}_K$. Without loss of generality we may assume that it is one-dimensional, i.e., that $P_0=<\cdot , \xi > \xi$, for an element $\xi \in H$ with $\Vert \xi \Vert =1$. Then

$$\begin{aligned} \Vert P\xi _n\Vert ^2&= \left(\sum _{i=n}^{2n} P_0\eta _i\otimes \gamma _i, \sum _{j=n}^{2n} P_0\eta _j\otimes \gamma _j\right) \\&= \left(\sum _{i=n}^{2n} <\eta _i, \xi > \xi \otimes \gamma _i, \sum _{j=n}^{2n} <\eta _j, \xi > \xi \otimes \gamma _j\right)\\&= \sum _{i,j=n}^{2n} \left(<\eta _i, \xi > \xi , \ <\eta _j, \xi > \xi \right)_H \cdot (\gamma _i, \gamma _j)_K = \sum _{k=n}^{2n} |<\eta _k, \xi >|^2. \end{aligned}$$

Since the series $\sum _{k=1}^\infty |<\eta _k, \xi >|^2$ converges, the sequence $\sum _{k=n}^{2n} |<\eta _k, \xi >|^2$ converges to $0$, when $n\rightarrow \infty$. Hence $\Vert P\xi _n\Vert \rightarrow 0$. Therefore $P\xi _n \stackrel{S}{\longrightarrow } P\xi =0$, but the sequence $\{\xi _n\}$ is unbounded. Moreover it is easy to show that the sequence $\{\eta _n\otimes \gamma \}$ $(RW)_r$ converges to $0$, but it does not converge strongly to $0$, and the sequence $\{\xi \otimes \gamma _n\}$ converges weakly to $0$, but does not $(RW)_r$ converge to $0$.

It is easy to see that the following assertions are true.

Lemma4

$$\begin{aligned} \xi _n \stackrel{RW}{\longrightarrow } \xi&\ \Longleftrightarrow&\ \xi _n \stackrel{(RW)_r}{\longrightarrow } \xi \end{aligned}$$

(1)

Here $\xi _n \stackrel{RW}{\longrightarrow } \xi$ means that $\Vert \xi _n\Vert$ is bounded and $P\xi _n \stackrel{S}{\longrightarrow } P\xi$, for every projection $P\in M$ (see [7, Definition 1]).

Proof

The proof of implication “$\Rightarrow$” is obvious. Let’s prove the implication “$\Leftarrow$”. Assume, that there is some projection $p\in M$ with $\Vert p\xi _n\Vert \not \rightarrow 0$. Since the projections $p$ and $\alpha (p)$ are equivalent we have $\Vert \alpha (p)\xi _n\Vert \not \rightarrow 0$ (see [1]). It is easy to see that $a=p+\alpha (p)\in (M,\alpha )$ and $a\ge p$ (because $\alpha (p)\ge 0$), therefore $\Vert a\xi _n\Vert \not \rightarrow 0$. Then exists a spectral projection $e\in (M,\alpha )$ of an element $a$ with $\Vert e\xi _n\Vert \not \rightarrow 0$. It is a contradiction with the assumption. $\square$

The following theorem is a generalization of Hibert’s characterization of the compact operators.

Theorem5

(Theorem 7, [7]) An element $A$ is compact in $M$ iff it maps (RW) converging sequences into strongly converging ones.

We have the following real analogue of the above characterization.

Theorem6

An element $A$ is compact in $(M,\alpha )$ iff it maps (RW)$_r$ converging sequences into strongly converging ones.

Proof

Firstly, let us prove the sufficiency. Let $I$ be the two-sided closed (pure real) ideal generated by the projections which are finite relative to a real W*-algebra $(M,\alpha )$ and put $J=I+iI$. As it was noticed above $J$ is the two-sided closed (complex) ideal, generated by the projections which are finite relative to a W*-algebra $M$. If we put

$$\begin{aligned} I_1 = \{A\in (M,\alpha ): \ A\xi _n \stackrel{S}{\rightarrow } 0, \ \mathrm{for \ any \ sequence} \ \{\xi _n\}\subset H_r \ \mathrm{with} \ \xi _n \stackrel{(RW)_r}{\longrightarrow } 0\}, \end{aligned}$$

then by (1) for

$$\begin{aligned} J_1 = \{A\in M: \ A\xi _n \stackrel{S}{\rightarrow } 0, \ \mathrm{for \ any \ sequence} \ \{x_n\}\subset H \ \mathrm{with} \ \xi _n \stackrel{(RW)}{\longrightarrow } 0\}, \end{aligned}$$

we obtain $I_1\subset J_1=J=I+iI$. Here the equality $J_1=J$ is valid by Theorem 5. Since $I_1\subset (M,\alpha )$, we obtain $I_1\subset I$. Therefore $A\in I$, i.e. $A$ is compact relatively to $(M,\alpha )$

Now we shall prove the necessity. It suffices to show that $I\subset I_1$. Suppose that $K\in I$ and $\xi _n \stackrel{(RW)_r}{\longrightarrow } \xi$. Without loss of generality we can assume that $\xi =\theta$. We repeat step by step the scheme of proof of Theorem 1.3 [8], to obtain the real analogue of the generalized Rellich condition for $K$, i.e., for every $\lambda >0$ there is a projection $P$ of $(M,\alpha )$ such that $\Vert KP\Vert \le \lambda$ and $\mathbf{1}\!\!\mathrm{I}- P$ is finite. From $\xi _n \stackrel{(RW)_r}{\longrightarrow } 0$, by definition, we obtain $(\mathbf{1}\!\!\mathrm{I}- P)\xi _n \stackrel{S}{\longrightarrow } 0$, and hence $K(\mathbf{1}\!\!\mathrm{I}- P)\xi _n \stackrel{S}{\longrightarrow } 0$. Since $\Vert K-K(\mathbf{1}\!\!\mathrm{I}- P)\Vert =\Vert KP\Vert \le \lambda$ and $\Vert \xi _n\Vert$ is bounded by definition, we have $K\xi _n \stackrel{S}{\longrightarrow } 0$. Thus, we have shown that for any $K\in I$ the condition $\xi _n \stackrel{(RW)_r}{\longrightarrow } \xi$ implies $K\xi _n \stackrel{S}{\longrightarrow } K\xi$. Hence $K\in I_1$, and therefore $I_1\subset I$. $\square$

Theorem 6 shows that Hibert’s characterization of the compact operators remains valid in semifinite real W*-algebras.