Abstract
We prove that if A is a non-separable abelian tracial von Neuman algebra then its free powers A∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group, \(\mathcal{F}(A^{*n})=1\), whenever 2≤n<∞. This settles the non-separable version of the free group factor problem.
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1 Introduction
The free group factor problem, asking whether the II1 factors \(L\mathbb{F}_{n}\) arising from the free groups with n generators \(\mathbb{F}_{n}\), 2≤n≤∞, are isomorphic or not, is perhaps the most famous in operator algebras, being in a way emblematic for this area, broadly known even outside of it.
It is generally believed that the free group factors are not isomorphic. Since \(L\mathbb{F}_{n}=L\mathbb{Z}*\cdots *L\mathbb{Z}\), this amounts to A∗n,2≤n≤∞, being non-isomorphic, where \(A=L\mathbb{Z}\) is the unique (up to isomorphism) separable diffuse abelian von Neumann algebra. Due to work in [Rad94, Dyk94], based on Voiculescu’s free probability methods, this is also equivalent to the fundamental group of A∗n being trivial for some (equivalently, all) 2≤n<∞, \(\mathcal {F}(A^{*n})=1\).
We study here the non-separable version of the free group factor problem, asking whether the II1 factors A∗n,2≤n≤∞, are non-isomorphic when A is an abelian but non-separable von Neumann algebra (always assumed tracial, i.e., endowed with a given normal faithful trace). Examples of such algebras A include the ultrapower von Neumann algebra \((L\mathbb{Z})^{\omega}\) and the group von Neumann algebra LH, where ω is a free ultrafilter on \(\mathbb{N}\) and H is an uncountable discrete abelian group, such as \(\mathbb{R}\) or \(\mathbb{Z}^{\omega}\). We obtain the following affirmative answer to the problem:
Theorem 1.1
Let A be a diffuse non-separable abelian tracial von Neumann algebra.
Then the II1 factors A∗n,2≤n≤∞, are mutually non-isomorphic, and have trivial fundamental group, \(\mathcal {F}(A^{*n})=1\), whenever 2≤n<∞.
In other words, if the abelian components of a free product A∗n are being “magnified” from separable to non-separable, then the corresponding II1 factors do indeed remember the number of terms involved. One should note that if 2≤n≤∞, then any II1 factor A∗n, with A diffuse abelian, is an inductive limit of subfactors isomorphic to \(L\mathbb{F}_{n}\).
To prove Theorem 1.1, we show that the II1 factors of the form M=A1∗⋯∗An, with A1,A2,…,An non-separable abelian, have a remarkably rigid structure. Specifically, we prove that given any unital abelian von Neumann subalgebra B⊂pMp that is purely non-separable (i.e., has no separable direct summand) and singular (i.e., has trivial normalizer), there is a partition of p into projections pi∈B such that Bpi is unitarily conjugate to a direct summand of Ai, for every 1≤i≤n (see Corollary 3.7). This implies that the family {Aipi}i, consisting of the maximal purely non-separable direct summands of Ai, 1≤i≤n, coincides with the sans-core of M, a term we use to denote the maximal family \(\mathcal {A}^{ns}_{M} = \{B_{j}\}_{j}\) of pairwise disjoint, singular, purely non-separable abelian subalgebras Bj of M. The uniqueness (up to unitary conjugacy, cutting and gluing) of this family ensures that the sans-rank of M, defined by
is an isomorphism invariant for M. This shows in particular that if A is a diffuse non-separable abelian von Neumann algebra and Ap is its maximal purely non-separable direct summand, then rns(A∗n)=nτ(p), for every 2≤n≤∞, implying the non-isomorphism in the first part of Theorem 1.1. Since the sans-rank is easily seen to satisfy the amplification formula rns(Mt)=rns(M)/t, for every t>0, the last part of the theorem follows as well.
We define the sans-core and sans-rank of a II1 factor in Sect. 2, where we also discuss some basic properties, including the amplification formula for the sans-rank. In Sect. 3 we prove that rns(∗i∈IMi)=∑i∈Irns(Mi), for any family Mi,i∈I, of tracial von Neumann algebras (see Theorem 3.8) and use this formula to deduce Theorem 1.1. The proof of Theorem 3.8 uses intertwining by bimodules techniques and control of relative commutants in amalgamated free product II1 factors from [IPP08]. Notably, we use results from [IPP08] to show that any von Neumann subalgebra P of a tracial free product M=M1∗M2 which has a non-separable relative commutant, P′∩M, must have a corner which embeds into M1 or M2 (see Theorem 3.4). The last section of the paper, Sect. 4, records some further remarks and open problems.
2 The singular abelian core of a II1 factor
The aim of this section is to define the singular abelian core a II1 factor and its non-separable analogue. We start by recalling some terminology involving von Neumann algebras. We will always work with tracial von Neumann algebras, i.e., von Neumann algebras M endowed with a fixed faithful normal trace τ. We endow M with the 2-norm given by ∥x∥2=τ(x∗x)1/2 and denote by \(\mathcal {U}(M)\) its group of unitaries and by (M)1={x∈M∣∥x∥≤1} its (uniform) unit ball. We assume that all von Neumann subalgebras are unital. For a von Neumann subalgebra A⊂M, we denote by EA:M→M the conditional expectation onto A and by \(\mathcal {N}_{M}(A)=\{u\in \mathcal {U}(M)\mid uAu^{*}=A\}\) the normalizer of A in M. We say that a von Neumann algebra M is purely non-separable if pMp is non-separable, for every nonzero projection p∈M.
2.1 Interwining by bimodules
We recall the intertwining by bimodules theory from [Pop06b, Theorem 2.1 and Corollary 2.3].
Theorem 2.1
[Pop06b]
Let (M,τ) be a tracial von Neumann algebra and A⊂pMp,B⊂qMq be von Neumann subalgebras. Then the following conditions are equivalent.
-
(1)
There exist nonzero projections p0∈A,q0∈B, a ∗-homomorphism θ:p0Ap0→q0Qq0 and a nonzero partial isometry v∈q0Mp0 such that θ(x)v=vx, for all x∈p0Ap0.
-
(2)
There is no net \(u_{n}\in \mathcal {U}(A)\) satisfying ∥EB(x∗uny)∥2→0, for all x,y∈pM.
If (1) or (2) hold true, we write A≺MB and say that a corner of A embeds into B inside M. If Ap′≺MB, for any nonzero projection p′∈A∩pMp, we write \(A\prec ^{\text{f}}_{M}B\).
2.2 Singular MASAs
Let (M,τ) be a tracial von Neumann algebra. An abelian von Neumann subalgebra A⊂M is called a MASA if it is maximal abelian and singular if it satisfies \(\mathcal {N}_{M}(A)=\mathcal {U}(A)\) [Dix54]. Note that a singular abelian von Neumann subalgebra A⊂M is automatically a MASA.
Two MASAs A⊂pMp,B⊂qMq are called disjoint if A⊀MB. The following result from [Pop06a, Theorem A.1] shows that disjointness for MASAs is the same as having no unitarily conjugated corners. In particular, disjointness of MASAs is a symmetric relation.
Theorem 2.2
[Pop06a]
Let (M,τ) be a tracial von Neumann algebra and A⊂pMp,B⊂qMq be MASAs. Then A≺MB if and only if B≺MA and if and only if there exist nonzero projections p0∈A,q0∈B such that u(Ap0)u∗=Bq0, for some \(u\in \mathcal {U}(M)\).
2.3 The singular abelian core
We are now ready to give the following:
Definition 2.3
Let (M,τ) be a tracial von Neumann algebra. We denote by \(\mathcal {S}(M)\) the set of all families \(\mathcal {A}=\{A_{i}\}_{i\in I}\), where pi∈M is a projection, Ai⊂piMpi is a singular MASA, for every i∈I, and Ai,Ai′ are disjoint, for every i,i′∈I with i≠i′. We denote \(\text{d}(\mathcal {A})=\sum _{i\in I}\tau (p_{i})\), the size of the family \(\mathcal{A}\). Given \(\mathcal {A}=\{A_{i}\}_{i\in I},\mathcal {B}=\{B_{j}\}_{j\in J}\in \mathcal {S}(M)\) we write \(\mathcal {A}\leq \mathcal {B}\) if for every i∈I and nonzero projection p∈Ai, there exists j∈J such that Aip≺MBj. We say that \(\mathcal {A}\) and \(\mathcal {B}\) are equivalent and write \(\mathcal {A}\sim \mathcal {B}\) if \(\mathcal {A}\leq \mathcal {B}\) and \(\mathcal {B}\leq \mathcal {A}\).
Lemma 2.4
Let (M,τ) be a tracial von Neumann algebra. Then \(\mathcal {S}(M)\) admits a maximal element with respect ≤. Moreover, any two maximal elements of \(\mathcal {S}(M)\) with respect to ≤ are equivalent.
Proof
Let \(\mathcal {A}=\{A_{i}\}_{i\in I}\in \mathcal {S}(M)\) be a maximal family with respect to inclusion. Then \(\mathcal {A}\) is maximal with respect to ≤. To see this, let \(\mathcal {B}=\{B_{j}\}_{j\in J}\in \mathcal {S}(M)\). If \(\mathcal {B}\nleq \mathcal {A}\), then there are j∈J and a nonzero projection q∈Bj with Bjq⊀MAi, for every i∈I. As Bjq⊂qMq is a singular MASA, we get that \(\mathcal {A}\cup \{B_{j}q\}\in \mathcal {S}(M)\), contradicting the maximality of \(\mathcal {A}\) with respect to inclusion. The moreover assertion follows. □
Definition 2.5
Let (M,τ) be a tracial von Neumann algebra. We denote by \(\mathcal {A}_{M}\) the equivalence class consisting of all maximal elements of \(\mathcal {S}(M)\) with respect to ≤, and call it the singular abelian core of M. We define the rank r(M) of M as the size, \(\text{d}(\mathcal {A})\), of any \(\mathcal {A}\in \mathcal {A}_{M}\). Note that r(M) is a well-defined isomorphism invariant of M since the map \(\mathcal {A}\mapsto \text{d}(\mathcal {A})\) is constant on equivalence classes.
Remark 2.6
Definition 2.3 presents the folded form of \(\mathcal {S}(M)\), for a tracial von Neumann algebra (M,τ). Let K be a large enough set, which contains the index set I of any element \(\mathcal {A}=\{A_{i}\}_{i\in I}\) of \(\mathcal {S}(M)\). For instance, take K to be the collection of all singular MASAs A⊂pMp, for all projections p∈M. We identify every \(\mathcal {A}=\{A_{i}\}_{i\in I}\) of \(\mathcal {S}(M)\) with the singular abelian von Neumann subalgebra \(\mathcal {A}=\oplus _{i\in I}A_{i}\) of \(p\mathcal {M}p\), where \(\mathcal {M}=M\overline{\otimes}\mathbb{B}(\ell ^{2}K)\) and \(p=\oplus _{i\in I}p_{i}\in \mathcal {M}\). This is the unfolded form of \(\mathcal {S}(M)\). In this unfolded form, given \(\mathcal {A},\mathcal {B}\in \mathcal {S}(M)\), we have that \(\mathcal {A}\leq \mathcal {B}\) (respectively, \(\mathcal {A}\sim \mathcal {B}\)) if and only if \(\mathcal {A}\subset u\mathcal {B}qu^{*}\) (respectively, \(\mathcal {A}=u\mathcal {B}u^{*}\)), for a projection \(q\in \mathcal {B}\) and unitary \(u\in \mathcal {M}\).
The unfolded form of the singular abelian core \(\mathcal {A}_{M}\) of M is then the unique (up to unitary conjugacy) singular abelian von Neumann subalgebra \(\mathcal {A}\subset p\mathcal {M}p\) generated by finite projections such that for any singular abelian von Neumann subalgebra \(\mathcal {B}\subset q\mathcal {M}q\), for a finite projection q, we have that \(\mathcal {B}\prec _{\mathcal {M}}\mathcal {A}\). The rank r(M) is then equal to the semifinite trace, (τ⊗Tr)(p), of the unit p of \(\mathcal {A}_{M}\). Notice that if the semifinite trace (τ⊗Tr)(p) of the support of \(\mathcal {A}\) is infinite, then it can be viewed as a cardinality ≤|K|. We will in fact view r(M) this way, when infinite.
Remark 2.7
Let M be an arbitrary separable II1 factor. By a result in [Pop83c], M admits a singular MASA. This result was strengthened in [Pop19, Theorem 1.1] where it was shown that M contains an uncountable family of pairwise disjoint singular MASAs. Consequently, r(M)>ℵ0. More recently, it was shown in [Pop21, Theorem 1.1] that M contains a copy of the hyperfinite II1 factor R⊂M which is coarse, i.e., such the R-bimodule L2(M)⊖L2(R) is a multiple of the coarse R-bimodule \(\text{L}^{2}(R)\overline{\otimes}\text{L}^{2}(R)\). In combination with [Pop21, Proposition 2.6.3] and [Pop14, Theorem 5.1.1], this implies that M has a continuous family of disjoint singular MASAs. Since the set of distinct self-adjoint elements in a separable II1 factor has continuous cardinality \(\mathfrak {c}=2^{\aleph _{0}}\) and each singular MASA is generated by a self-adjoint element, it follows that \(\text{r}(M)=\mathfrak {c}\), for every separable II1 factor M.
2.4 The singular abelian non-separable core
Remark 2.7 shows that the rank r(M) is equal to the continuous cardinality \(\mathfrak {c}\) for any separable II1 factor M, and thus cannot be used to distinguish such factors up to isomorphism. In contrast, we define in this section a non-separable analogue of r(M), which will later enable us to prove the non-isomorphisms asserted by Theorem 1.1.
Definition 2.8
Let (M,τ) be a tracial von Neumann algebra. We say that a von Neumann subalgebra A⊂pMp is a sans-subalgebra of M if it is singular abelian in pMp and purely non-separable. We denote by \(\mathcal {S}_{\text{ns}}(M)\subset \mathcal {S}(M)\) the set of \(\mathcal {A}=\{A_{i}\}_{i\in I}\in \mathcal {S}(M)\) such that Ai is a sans-subalgebra, for every i∈I. We call any \(\mathcal {A}\in \mathcal{S}_{\text{ns}}(M)\) a sans family in M.
Since Lemma 2.4 trivially holds true if we replace \(\mathcal {S}(M)\) by \(\mathcal {S}_{\text{ns}}(M)\), we can further define:
Definition 2.9
Let (M,τ) be a tracial von Neumann algebra. We denote by \(\mathcal {A}_{M}^{\text{ns}}\) the equivalence class consisting of all maximal elements of \(\mathcal {S}_{\text{ns}}(M)\) with respect to ≤, and call it the singular abelian non-separable core (abbreviated, the sans-core) of M. We define the sans-rank rns(M) of M as the size, \(\text{d}(\mathcal {A})\), of any \(\mathcal {A}\in \mathcal {A}_{M}^{\text{ns}}\).
Remark 2.10
Like in Remark 2.6, consider \(\mathcal {M}=M\overline{\otimes}\mathbb{B}(\ell ^{2}K)\), for a large enough set K. In the unfolded form of \(\mathcal {S}_{\text{ns}}(M)\), the sans-core \(\mathcal {A}_{M}^{\text{ns}}\) of M is the unique (up to unitary conjugacy) sans-subalgebra \(\mathcal {A}\subset p\mathcal {M}p\) generated by finite projections such that for any sans-subalgebra \(\mathcal {B}\subset q\mathcal {M}q\), for a finite projection q, we have that \(\mathcal {B}\prec _{\mathcal {M}}\mathcal {A}\). The sans-rank rns(M) is then the semifinite trace, (τ⊗Tr)(p), of the unit p of \(\mathcal {A}_{M}^{\text{ns}}\). Like in Remark 2.6, when the semifinite trace of the support of the sans-core in this unfolded form is infinite, then we will view rns(M) as a cardinality ≤|K|.
Remark 2.11
If M is a separable II1 factor, then we clearly have rns(M)=0. If A⊂M is a singular MASA and ω is a free ultrafilter on \(\mathbb{N}\), then Aω⊂Mω is a purely non-separable singular MASA, see [Pop83c, 5.3]. Moreover, disjoint MASAs in M give rise to disjoint ultrapower MASAs in Mω. By using these facts and results from [Pop14, Pop21] as in Remark 2.7 we get that \(\text{r}_{\text{ns}}(M^{\omega})\geq \mathfrak {c}\), for every separable II1 factor M. But getting \(\text{r}_{\text{ns}}(M^{\omega})\leq \mathfrak {c}\) is problematic, as besides the family of disjoint ultraproduct singular MASAs in Mω, which has cardinality \(\mathfrak {c}\), one may have singular MASAs that are not of this form.
The expression of rns(M) as the semifinite trace of the support of the sans-core in unfolded form, as in Remark 2.10, implies the following scaling formula for rns(M). We include below an alternative short proof using the folded form of \(\mathcal {S}_{\text{ns}}(M)\).
Proposition 2.12
Let M be any II1 factor and \(t \in \mathbb{R}_{+}^{*}\). Then we have
In particular, if 0<rns(M)<∞, then M has trivial fundamental group, \(\mathcal {F}(M)=\{1\}\).
Proof
It is enough to argue that rns(qMq)=rns(M)/τ(q), for every nonzero projection q∈M. This follows immediately by using the fact that any \(\mathcal {A}=\{A_{i}\}_{i\in I}\in \mathcal {S}(M)\) is equivalent to some \(\mathcal {B}=\{B_{j}\}_{j\in J}\in \mathcal {S}(M)\), such that Bj⊂qjMqj, for some qj≤q, for every j∈J. □
3 Main results
3.1 Main technical result
This subsection is devoted to proving our main technical result. Throughout the subsection we use the following notation. Let (M1,τ1) and (M2,τ2) tracial von Neumann algebras and denote by M=M1∗M2 their free product with its canonical trace τ.
Theorem 3.1
Let P⊂pMp be a von Neumann subalgebra such that P′∩pMp is non-separable. Then P≺MM1 or P≺MM2.
The proof of Theorem 3.1 is based on the main technical result of [IPP08]. By [PV10, Sect. 5.1], given ρ∈(0,1), we have a unital tracial completely positive map mρ:M→M such that mρ(x1x2⋯xn)=ρnx1x2⋯xn, for every \(n\in \mathbb{N}\) and \(x_{i}\in M_{i_{j}}\ominus \mathbb{C}1\), where ij∈{1,2}, for every 1≤j≤n, and ij≠ij+1, for every 1≤j≤n−1. Note that \(\lim \limits _{\rho \rightarrow 1}\|\text{m}_{\rho}(x)-x\|_{2}=0\) and the map (0,1)∋ρ↦∥mρ(x)∥2 is increasing, for every x∈M. The implication (1) ⇒ (2) follows from [IPP08, Theorem 4.3], formulated here as in [PV10, Theorem 5.4], see also [Hou09, Sect. 5].
Theorem 3.2
[IPP08]
Let P⊂pMp be a von Neumann subalgebra. Then the following two conditions are equivalent:
-
(1)
There exists ρ∈(0,1) such that \(\inf _{u\in \mathcal {U}(P)}\|\mathrm{m}_{\rho}(u)\|_{2}>0\).
-
(2)
P≺MM1 or P≺MM2.
Proof
Assume that (1) holds. Since \(\tau (x^{*}\text{m}_{\rho ^{2}}(x))=\|\text{m}_{\rho}(x)\|_{2}^{2}\), for every x∈M, we get that \(\inf _{u\in \mathcal {U}(P)}\tau (u^{*}\text{m}_{\rho ^{2}}(u))>0\) and [PV10, Theorem 5.4] implies (2).
To see that (2) ⇒ (1), assume that P≺MMi, for some i∈{1,2}. By Theorem 2.1 we find a nonzero partial isometry v∈M such that v∗v=p0p′, for some projections p0∈P,p′∈P′∩pMp, and (p0Pp0)1p′⊂v∗(Mi)1v. Since ∥mρ(x)−x∥2≤|ρ−1|, for every x∈(Mi)1, we get that \(\lim _{\rho \rightarrow 1}(\sup _{x\in (p_{0}Pp_{0})_{1}p'}\| \text{m}_{\rho}(x)-x\|_{2})=0\). Let p1 be the central support of p0 in P and denote p″=p1p′∈P′∩pMp. It follows that \(\lim _{\rho \rightarrow 1}(\sup _{x\in (Pp'')_{1}}\|\text{m}_{\rho}(x)-x \|_{2})=0\). From this it is easy to deduce that \(\liminf _{\rho \rightarrow 1}(\inf _{u\in \mathcal {U}(P)}\|\text{m}_{ \rho}(u)\|_{2})\geq \|p''\|_{2}>0\), which clearly implies (1). □
Corollary 3.3
Let P⊂pMp be a von Neumann subalgebra such that P⊀MM1 and P⊀MM2. Then there exists a separable von Neumann subalgebra Q⊂P such that Q⊀MM1 and Q⊀MM2.
Proof
Since P⊀MM1 and P⊀MM2, by Theorem 3.2 we find a sequence \(u_{n}\in \mathcal {U}(P)\) such that ∥m1−1/n(un)∥2≤1/n. Let Q⊂P be the separable von Neumann subalgebra generated by {un}n≥1. Let ρ∈(0,1). Then for every n≥1 such that ρ≤1−1/n we have that ∥mρ(un)∥2≤∥m1−1/n(un)∥2≤1/n. This implies \(\inf _{u\in \mathcal {U}(Q)}\|\text{m}_{\rho}(u)\|_{2}=0\). Since this holds for every ρ∈(0,1), Theorem 3.2 implies that Q⊀MM1 and Q⊀MM2. □
Lemma 3.4
Let Q⊂M be a separable von Neumann subalgebra. Then we can find separable von Neumann subalgebras N1⊂M1 and N2⊂M2 such that Q⊂N1∗N2.
Proof
For i∈{1,2} let \(\mathcal {B}_{i}\) be an orthonormal basis of \(\text{L}^{2}(M_{i})\ominus \mathbb{C}1\) such that \(\mathcal {B}_{i}\subset M_{i}\ominus \mathbb{C}1\). Let \(\mathcal {B}_{0}\) be the set of ξ1ξ2⋯ξn, where \(n\in \mathbb{N}\), \(\xi _{i}\in \mathcal {B}_{i_{j}}\), for some ij∈{1,2}, for every 1≤j≤n, and ij≠ij+1, for every 1≤j≤n−1. Then \(\mathcal {B}=\mathcal {B}_{0}\cup \{1\}\) is an orthonormal basis of L2(M).
Let {xk}k≥1 be a sequence which generates Q. Then \(\mathcal {C}=\cup _{k\geq 1}\{\xi \in \mathcal {B}\mid \langle x_{k}, \xi \rangle \neq0\}\) is countable. For i∈{1,2}, let \(\mathcal {C}_{i}\) be the countable set of all \(\xi \in \mathcal {B}_{i}\) which appear in the decomposition of some element of \(\mathcal {C}\). The von Neumann subalgebra Ni of Mi generated by \(\mathcal {C}_{i}\) is separable, for every i∈{1,2}. Since by construction we have that Q⊂N1∗N2, this finishes the proof. □
Proof of Theorem 3.1
Assume by contradiction that P⊀MM1 and P⊀MM2. By applying Corollary 3.3, we can find a separable von Neumann subalgebra Q⊂P such that Q⊀MM1 and Q⊀MM2. By Lemma 3.4, we can further find separable von Neumann subalgebras N1⊂M1 and N2⊂M2, such that Q⊂N:=N1∗N2. Denote R=M1∗N2.
Since Q⊀MM1, Q⊂R⊂M and N1⊂M1, we get that Q⊀RN1. Since Q⊂N and \(R=M_{1}*_{N_{1}}N\), [IPP08, Theorem 1.1] implies that Q′∩R=Q′∩N. Next, since Q⊀MM2 and N2⊂M2, we get that Q⊀MN2. Since Q⊂R and \(M=R*_{N_{2}}M_{2}\), applying [IPP08, Theorem 1.1] again gives that Q′∩M=Q′∩R. Altogether, we get that Q′∩M=Q′∩N. Since N and thus Q′∩N is separable, using that P′∩M⊂Q′∩M, we conclude that P′∩M is separable. □
3.2 Non-separable MASAs in free product algebras
In this subsection, we derive some consequences of Theorem 3.1 to the structure of non-separable MASAs in free product algebras.
Corollary 3.5
Let (M1,τ1) and (M2,τ2) be tracial von Neumann algebras, and denote by M=M1∗M2 their free product. Let A⊂pMp be a purely non-separable MASA. Then there exist projections (pk)k∈K⊂A and unitaries (uk)k∈K⊂M such that ∑k∈Kpk=p and for every k∈K, \(u_{k}Ap_{k}u_{k}^{*}\subset M_{i}\), for some i∈{1,2}.
Proof
By a maximality argument, it suffices to prove that if q∈A is a nonzero projection, then there are a nonzero projection r∈Aq, a unitary u∈M and i∈{1,2} such that uAru∗⊂Mi.
To this end, let q∈A be a nonzero projection. Since (Aq)′∩qMq=Aq is non-separable, Theorem 3.1 implies that there is i∈{1,2} such that Aq≺MMi. By Theorem 2.1, we can find nonzero projections e∈Aq,f∈Mi, a nonzero partial isometry v∈fMe and a ∗-homomorphism θ:Ae→fMif such that θ(x)v=vx, for every x∈Ae. Then r:=v∗v∈(Ae)′∩eMe=Ae and vv∗∈θ(Ae)′∩fMf. Since θ(Ae)⊂fMif is diffuse, by applying [IPP08, Theorem 1.1] (see also [Pop83b, Remarks 6.3.2)]) we get that vv∗∈fMif. Finally, let u∈M be any unitary such that ur=v. Then uAru∗=vArv∗=vAev∗=θ(Ae)vv∗⊂Mi, which finishes the proof. □
We continue by generalizing Corollary 3.5 to arbitrary tracial free products.
Corollary 3.6
Let (Mi,τi), i∈I, be a collection of tracial von Neumann algebras, and denote by M=∗i∈IMi their free product. Let A⊂pMp be a purely non-separable MASA. Then there exist projections (pk)k∈K⊂A and unitaries (uk)k∈K⊂M such that ∑k∈Kpk=p and for every k∈K, \(u_{k}Ap_{k}u_{k}^{*}\subset M_{i}\), for some i∈I.
Proof
Let A0⊂A be a separable diffuse von Neumann subalgebra. Reasoning similarly to the proof of Lemma 3.4 yields a countable set J⊂I such that A0⊂∗j∈JMj. Since A0 is diffuse, [IPP08, Theorem 1.1] gives that \(A\subset A_{0}'\cap pMp\subset *_{j\in J}M_{j}\). Thus, in order to prove the conclusion, after replacing I with J, we may take I countable. Enumerate I={im}m≥1.
Let {pk}k∈K⊂A be a maximal family, with respect to inclusion, of pairwise orthogonal projections such that for every k∈K, there are a unitary uk∈M and i∈I such that \(u_{k}Ap_{k}u_{k}^{*}\subset M_{i}\). In order to prove the conclusion it suffices to argue that ∑k∈Kpk=p. Put r:=p−(∑k∈Kpk).
Assume by contradiction that r≠0. We claim that
Otherwise, if (3.1) fails for some n≥1, then the proof of Corollary 3.5 gives a nonzero projection s∈Ar and a unitary u∈M such that \(uAsu^{*}\subset *_{m\leq n}M_{i_{m}}\). Applying Corollary 3.5 repeatedly gives a nonzero projection t∈As and a unitary \(v\in *_{m\leq n}M_{i_{m}}\) such that \(vuAtu^{*}v^{*}\subset M_{i_{m}}\), for some 1≤m≤n. This contradicts the maximality of the family {pk}k∈K, and proves (3.1).
If e∈(Ar)′∩rMr=Ar is a nonzero projection, then (Ae)′∩eMe=Ae is nonseparable. Since \(Ae\nprec _{M}*_{m\leq n}M_{i_{m}}\) by (3.1), Theorem 3.4 implies that \(Ae\prec _{M}*_{m>n}M_{i_{m}}\) and thus
To get a contradiction, we follow the proof of [HU16, Proposition 4.2]. Let \(\widetilde{M}=M*M\), identify M with \(M*1\subset \widetilde{M}\), and denote by θ the free flip automorphism of \(\widetilde{M}\). Endow \(\mathcal {H}=\text{L}^{2}(\widetilde{M})\) with the M-bimodule structure given by x⋅ξ⋅y=θ(x)ξy, for every x,y∈M and \(\xi \in \mathcal {H}\). Using (3.2), the proof of [HU16, Proposition 4.2] yields a sequence of vectors \(\eta _{n}\in r\cdot \mathcal {H}\cdot r\) such that ∥ηn∥2→∥r∥2, ∥x⋅ηn∥2≤∥x∥2 and ∥a⋅ηn−ηn⋅a∥2→0, for every x∈rMr and a∈Ar.
Next, we note that the Ar-bimodule \(r\cdot \mathcal {H}\cdot r\) is isomorphic to a multiple of the coarse Ar-bimodule, ⊕S(L2(Ar)⊗L2(Ar)), for some (possibly uncountable) set S. If ζ∈⊕S(L2(Ar)⊗L2(Ar)), then we can find a countable subset T⊂S such that ζ⊕T(L2(Ar)⊗L2(Ar)). By combining these two facts with the previous paragraph, we obtain a sequence of vectors \(\zeta _{n}\in \oplus _{\mathbb{N}}(\text{L}^{2}(Ar)\otimes \text{L}^{2}(Ar))\) such that ∥ζn∥2→∥r∥2, ∥a⋅ζn∥2≤∥a∥2 and ∥a⋅ζn−ζn⋅a∥2→0, for every a∈Ar. By reasoning similarly to the proof of Lemma 3.4, we find a separable von Neumann subalgebra A0⊂Ar such that \(\zeta _{n}\in \oplus _{\mathbb{N}}(\text{L}^{2}(A_{0})\otimes \text{L}^{2}(A_{0}))\).
As A0 is separable and Ar is purely non-separable, we derive that Ar⊀ArA0. Theorem 2.1 gives a unitary u∈Ar with \(\|\text{E}_{A_{0}}(u)\|_{2}\leq \|r\|_{2}/2\). Put \(a=u-\text{E}_{A_{0}}(u)\in A\). Since \(a\cdot \zeta _{n}\in \oplus _{\mathbb{N}}((\text{L}^{2}(Ar)\ominus \text{L}^{2}(A_{0}))\otimes \text{L}^{2}(A_{0}))\) and \(\zeta _{n}\cdot a\in \oplus _{\mathbb{N}} (\text{L}^{2}(A_{0})\otimes ( \text{L}^{2}(Ar)\ominus \text{L}^{2}(A_{0}))\), we have that 〈a⋅ζn,ζn⋅a〉=0, for every n. Using that ∥a⋅ζn−ζn⋅a∥2→0, we get that ∥a⋅ζn∥2→0. On the other hand, \(\|a\cdot \zeta _{n}\|_{2}\geq \|u\cdot \zeta _{n}\|_{2}-\|\text{E}_{A_{0}}(u) \cdot \zeta _{n}\|_{2}\geq \|\zeta _{n}\|_{2}-\|\text{E}_{A_{0}}(u)\|_{2} \geq \|\zeta _{n}\|_{2}-\|r\|_{2}/2\). Since ∥ζn∥2→∥r∥2>0, we altogether get a contradiction, which finishes the proof. □
We end this subsection by noticing that in the case A⊂pMp is a singular MASA and Mi is abelian, for every i∈I, the conclusion of Corollay 3.6 can be strengthened as follows:
Corollary 3.7
In the context of Corollary 3.6, assume additionally that A⊂pMp is singular and Mi is abelian, for every i∈I. Then there exist projections (qi)i∈I⊂A and unitaries (vi)i∈I⊂M such that ∑i∈Iqi=p, \(e_{i}=v_{i}q_{i}v_{i}^{*}\in M_{i}\) and \(v_{i}Aq_{i}v_{i}^{*}= M_{i}e_{i}\), for every i∈I.
Proof
By applying Corollary 3.6 we find projections (pk)k∈K⊂A and unitaries (uk)k∈K⊂M such that ∑k∈Kpk=p and for every k∈K, \(u_{k}Ap_{k}u_{k}^{*}\subset M_{i_{k}}\), for some ik∈I. Let k∈K and put \(r_{k}:=u_{k}p_{k}u_{k}^{*}\in M_{i_{k}}\). Since \(u_{k}Ap_{k}u_{k}^{*}\subset r_{k}Mr_{k}\) is a MASA and \(M_{i_{k}}\) is abelian we deduce that \(u_{k}Ap_{k}u_{k}^{*}=M_{i_{k}}r_{k}\), for every k∈K. Let k,k′∈K such that k≠k′ and ik=ik′. Since A⊂pMp is singular and pkpk′=0, there are no nonzero projections s∈Apk,s′∈Apk′ such that As and As′ are unitarily conjugated in M. This implies that rkrk′=0. Using this fact, it follows that if we denote \(q_{i}=\sum _{k\in K,i_{k}=i}p_{k}\), then \(v_{i}Aq_{i}v_{i}^{*}\subset M_{i}\), for every i∈I. For i∈I, let \(e_{i}=v_{i}q_{i}v_{i}^{*}\in M_{i}\). Then \(v_{i}Aq_{i}v_{i}^{*}\subset M_{i}e_{i}\) and since \(v_{i}Aq_{i}v_{i}^{*}\subset M_{i}e_{i}\) is a MASA, while Miei is abelian, it follows that \(v_{i}Aq_{i}v_{i}^{*}=M_{i}e_{i}\), as claimed. □
3.3 The non-separable rank of free product von Neumann algebras
In this section, we show that the sans core of a free product of tracial von Neumann algebras M=∗i∈IMi is the union of the sans cores of Mi,i∈I. This allows us to deduce that the sans rank of M is the sum of the sans ranks of Mi,i∈I.
Theorem 3.8
Let (Mi,τi), i∈I, be a colection of tracial von Neumann algebras, and denote by M=∗i∈IMi their free product. Then rns(M)=∑i∈Irns(Mi). Moreover, if \(\mathcal {A}_{i}\in \mathcal {A}_{M_{i}}^{\mathrm{ns}}\), for every i∈I, then \(\cup _{i\in I}\mathcal {A}_{i}\in \mathcal {A}_{M}^{\mathrm{ns}}\).
The moreover assertion uses implicitly the fact, explained in the proof, that every sans family in Mi is naturally a sans family in M, for every i∈I.
Proof
We have two inequalities to prove.
Inequality 1. rns(M)≥∑i∈Irns(Mi).
This inequality relies on several facts on free products, all of which follow from [IPP08, Theorem 1.1]. Let i,j∈I with i≠j.
-
(1)
If A⊂pMip is a MASA, then A⊂pMp is a MASA.
-
(2)
If A⊂pMip is a singular diffuse von Neumann subalgebra, then A⊂pMp is singular.
-
(3)
If A⊂pMip, B⊂qMiq are von Neumann subalgebras with A≺MB, then \(A \prec _{M_{i}} B\).
-
(4)
If A⊂pMip and B⊂qMjq are diffuse von Neumann subalgebras, then A⊀MB.
For i∈I, let \(\mathcal {A}_{i}\in \mathcal {A}_{M_{i}}^{\text{ns}}\) be a maximal sans family in Mi. We view every (not necessarily unital) subalgebra of Mi as a subalgebra of M. Then facts (1)-(3) imply that \(\mathcal {A}_{i}\) is a sans family in M. Moreover, fact (4) implies that \(\mathcal {A}:=\cup _{i\in I}\mathcal {A}_{i}\) is a sans family in M. Thus,
Inequality 2. rns(M)≤∑i∈Irns(Mi).
Let \(\mathcal{A}=\{A_{l}\}_{l\in L} \in \mathcal {A}^{\text{ns}}_{M}\) be a maximal sans family in M. Let l∈L. Applying Corollary 3.6 to Al gives projections \((p_{k,l})_{k\in K_{l}}\) and unitaries \((u_{k,l})_{k\in K_{l}}\) such that for every k∈Kl we have \(u_{k,l}A_{l}p_{k,l}u_{k,l}^{*}\subset M_{i}\), for some i∈I. For i∈I, let \(\mathcal {A}_{i}\in \mathcal {S}_{\text{ns}}(M_{i})\) be the collection of sans-subalgebras of Mi of the form \(u_{k,l}A_{l}p_{k,l}u_{k,l}^{*}\), for all l∈L,k∈Kl such that \(u_{k,l}A_{l}p_{k,l}u_{k,l}^{*}\subset M_{i}\). Then \(\mathcal {A}\) is equivalent to \(\cup _{i\in I}\mathcal {A}_{i}\), which allows us to conclude that
This finishes the proof of the main assertion. The moreover assertion now follows by combining the proofs of inequalities 1 and 2. □
3.4 Proof of Theorem 1.1
In preparation for the proof of Theorem 1.1, we first record the following direct consequence of Theorem 3.8:
Corollary 3.9
Let (Ai,τi), i∈I, be a collection of diffuse tracial abelian von Neumann algebras, and denote by M=∗i∈IAi their free product. For i∈I, let pi∈Ai be the maximal (possibly zero) projection such that Aipi is purely non-separable. Then rns(M)=∑i∈Iτi(pi). Moreover, if |I|≥2 and ∑i∈Iτi(pi)∈(0,+∞), then M is a II1 factor with \(\mathcal {F}(M)=\{1\}\). Also, the sans-core of M is given by \(\mathcal {A}_{M}^{ns}=\{A_{i}p_{i}\}_{i\in I}\).
Proof
Let i∈I. Since \(\{A_{i}p_{i}\}\in \mathcal {S}_{\text{ns}}(A_{i})\) is a maximal element, we get that rns(Ai)=τi(pi). The assertions now follow by using Theorem 3.8, Proposition 2.12, and the fact that any free product of diffuse tracial von Neumann algebras is a II1 factor. □
Proof of Theorem 1.1
Let (A,τ) be a diffuse non-separable tracial abelian von Neumann algebra. Let p∈A be the maximal, necessarily non-zero, projection such that Ap is purely non-separable. By Corollary 3.9, rns(A∗n)=nτ(p), for every 2≤n≤∞. Since p≠0, we get that A∗n, 2≤n≤∞, are mutually non-isomorphic, and \(\mathcal {F}(A^{*n})=\{1\}\), for 2≤n<∞. □
4 Further remarks and open problems
4.1 Freely complemented maximal amenable MASAs in A ∗n
The question of whether the II1 factors A∗n, 2≤n≤∞, are non-isomorphic for a non-separable diffuse tracial abelian von Neumann algebra A was asked in [BP]. This was motivated by the consideration of certain “radial-like” von Neumann subalgebras of M=A∗n, for 2≤n≤∞. Specifically, for every 1≤k≤n, let sk be a semicircular self-adjoint element belonging to Ak, the kth copy of A in M. For an ℓ2-summable family of real numbers t=(tk) with at least two non-zero entries, denote by A(t) the abelian von Neumann subalgebra of M generated by ∑ktksk. It was shown in [BP] that A(t)⊂M is maximal amenable and A(t),A(t′) are disjoint if t and t′ are not proportional. A key point in proving this result was to show that A(t)⊀MAk, for every k. Since the MASAs A(t) are separable, despite A being non-separable, this suggested that the only way to obtain a purely non-separable MASA in M is to “re-pack” pieces of Ak, 1≤k≤n. This further suggested the possibility of recovering n from the isomorphism class of M.
The construction of the family of radial-like maximal amenable MASAs A(t)⊂M in [BP] was triggered by an effort to obtain examples of non freely complemented maximal amenable MASAs in the free group factors \(L\mathbb{F}_{n}\). However, this remained open (see though [BP, Remark 1.4] for further comments concerning the inclusions A(t)⊂A∗n). Thus, there are no known examples of non freely complemented maximal amenable von Neumann subalgebras of \(L\mathbb{F}_{n}\). It may be that in fact any maximal amenable \(B\subset L\mathbb{F}_{n}\) is freely complemented (a property/question which we abbreviate as FC), see [Pop21, Question 5.5] and the introduction of [BP].
A test case for the FC question is proposed in the last paragraph of [Pop21]. There it is pointed out that if {Bi}i are diffuse amenable von Neumann subalgebras of \(L\mathbb{F}_{n}\) with Bi freely complemented and \(B_{i}\nprec _{L\mathbb{F}_{n}}B_{j}\), for every i≠j, then \(B=\oplus _{i} u_{i}p_{i}B_{i}p_{i}u_{i}^{*}\) is maximal amenable in M by [Pop83a], for any projections pi∈Bi and unitaries ui∈M satisfying \(\sum _{i} u_{i}p_{i}u_{i}^{*}=1\). Thus, if FC is to hold then B should be freely complemented as well.
The FC question is equally interesting for the factors M=A∗n with A purely non-separable abelian. If Ak denotes the kth copy of A in M, for every 1≤k≤n, then by Theorem 3.8, any purely non-separable singular abelian B⊂M is of the form \(B=\sum _{k} u_{k} A_{k}p_{k} u_{k}^{*}\) for some projections pk∈Ak and unitaries uk∈M with \(\sum _{k} u_{k}p_{k}u_{k}^{*}=1\). Thus, B is maximal amenable by [Pop83a]. Hence, if FC is to hold, then Theorem 3.8 suggests that the free complement of B could be obtained by a “free reassembling” of unitary conjugates of pieces of \(\{A_{k}(1-p_{k})\}_{k=1}^{n}\).
4.2 On the calculation of symmetries of A ∗n
Let M=A∗n with A purely non-separable abelian. Theorem 3.8 shows that if θ∈Aut(M) then \(\theta (\mathcal {A}^{\mathrm{ns}}_{M})=\mathcal {A}^{\mathrm{ns}}_{M}\), modulo the equivalence in \(\mathcal {S}_{\mathrm{ns}}(M)\) defined in Sect. 2.4. This suggests that one could perhaps explicitly calculate Out(M), for instance by identifying it with the Tr-preserving automorphisms α of the sans-core \(\mathcal {A}^{\mathrm{ns}}_{M}\), viewed in its unfolded form. In order to obtain from an arbitrary such α an automorphism θα of M it would be sufficient to solve the FC question in its “free repacking” form explained in Remark 4.1 above. To prove that such a map α↦θα is surjective one would need to show that if θ∈Aut(M) implements the identity on the sans-core \(\mathcal {A}^{\mathrm{ns}}_{M}\), then θ is inner on M.
This heuristic is supported by the case of automorphisms θ of the free group \(\mathbb{F}_{2}\): if θ(a)=a and θ(b)=gbg−1, for some \(g\in \mathbb{F}_{2}\), where a,b denote the free generators of \(\mathbb{F}_{2}\), then g must be of the form g=ak, and so θ=Ad(g) is inner.
However, this phenomenon fails for the free groups \(\mathbb{F}_{n}\) on n≥3 generators. Specifically, any \(e\neq g\in \mathbb{F}_{n-1}=\langle a_{1},\ldots ,a_{n-1}\rangle \) gives rise to an outer automorphism θg on \(\mathbb{F}_{n}\) defined by θg(ai)=ai, if 1≤i≤n−1, and θg(an)=gang−1, where a1,…,an are the free generators of \(\mathbb{F}_{n}\). Similarly, if M=A1∗⋯∗An, with Ai abelian diffuse, and n≥3, then any non-scalar unitary u∈A1∗⋯∗An−1∗1 gives rise to an outer automorphism θu of M defined by θu(x)=x, if x∈A1∗⋯∗An−1∗1, and θu(x)=uxu∗, if x∈1∗An.
A related problem is to investigate the structure of irreducible subfactors of finite Jones index N⊂M=A∗n, for A purely non-separable abelian, with an identification of the sans-core, the sans-rank of N and of the set of possible indices [M:N], in the spirit of [Pop06a, Sect. 7].
4.3 Amplifications of A ∗n
While Theorem 1.1 shows that \(\mathcal {F}(A^{*n})=1\) if A is non-separable abelian and n≥2 is finite, it is still of interest to identify the amplifications (A∗n)t, for t>0. For arbitrary t this remains open, but for t=1/k, \(k\in \mathbb{N}\), we have the following result. We are very grateful to Dima Shlyakhtenko for pointing out to us that the 1/2-amplification of A∗n can be explicitly calculated for arbitrary diffuse A by using existing models in free probability, a fact that stimulated us to investigate the general 1/k case.
Proposition 4.1
Let (Ai,τi), i∈I, be a countable collection of diffuse tracial abelian von Neumann algebras. Put M=∗i∈IAi and assume that |I|≥2. Let k≥2 and for every i∈I, let pi,1,…,pi,k∈Ai be projections such that τ(pi,j)=1/k, for every 1≤j≤k, and \(\sum _{j=1}^{k}p_{i,j}=1\).
Then M is a II1 factor and M1/k≅(∗i∈I,1≤j≤kAipi,j)∗D, where
-
(1)
\(D=L\mathbb{F}_{1+|I|k(k-1)-k^{2}}\), if I is finite, and
-
(2)
\(D=\mathbb{C}1\), if I is infinite.
Recall that the interpolated free group factors, \(L\mathbb{F}_{r}\), 1<r≤∞, introduced in [Rad94, Dyk94], satisfy the formulas
Proof
We will use the following consequence of [Dyk93, Theorem 1.2]:
Fact 4.2
[Dyk93]
Let P,Q be two tracial von Neumann algebras, and e∈P be a central projection (hence, P=Pe⊕P(1−e)). Denote R=P∗Q and \(S=(\mathbb{C}e\oplus P(1-e))*Q\subset R\). Then Pe and eSe are free and together generate eRe, hence eRe≅Pe∗eSe.
Specifically, we will use the following consequence of Fact 4.2:
Claim 4.3
Let P,Q be tracial von Neumann algebras and k≥2. Assume that P and Q admit projections e1,…,ek∈P and f1,…,fk∈Q such that ei is central in P, τ(ei)=τ(fi)=1/k, for every 1≤i≤k, \(\sum _{j=1}^{k}e_{j}=1\) and \(\sum _{j=1}^{k}f_{j}=1\). Then \(e_{1}(P*Q)e_{1}\cong Pe_{1}*\cdots *Pe_{k}*e_{1}((\mathbb{C}e_{1} \oplus \cdots \oplus \mathbb{C}e_{k})*Q)e_{1}\).
Proof of Claim 4.3
Note that e1 is equivalent to ej in \((\mathbb{C}e_{1}\oplus \cdots \oplus \mathbb{C}e_{k})*(\mathbb{C}f_{1} \oplus \cdots \oplus \mathbb{C}f_{k})\) and so in \((\mathbb{C}e_{1}\oplus \cdots \oplus \mathbb{C}e_{k})*Q\), for every 2≤j≤k. This follows from [Dyk94, Remark 3.3] if k=2 and because \((\mathbb{C}e_{1}\oplus \cdots \oplus \mathbb{C}e_{k})*(\mathbb{C}f_{1} \oplus \cdots \oplus \mathbb{C}f_{k})\cong \text{L}(\mathbb{Z}/k \mathbb{Z}*\mathbb{Z}/k\mathbb{Z})\) is a II1 factor if k≥3.
Denote \(e_{j}'=1-\sum _{l=1}^{j}e_{l}\) and \(P_{j}=\mathbb{C}e_{1}\oplus \cdots \oplus \mathbb{C}e_{j}\oplus Pe_{j}'\), for every 1≤j≤k. We claim that
When j=1, \(e_{1}'=1-e_{1}\) and thus equation (4.2) follows from Fact 4.2. Assume that (4.2) holds for some 1≤j≤k−1. Since ej+1∈Pj is a central projection, Pjej+1=Pej+1 and \(\mathbb{C}e_{j+1}\oplus P_{j}(1-e_{j+1})=P_{j+1}\), Fact 4.2 gives that ej+1(Pj∗Q)ej+1≅Pej+1∗ej+1(Pj+1∗Q)ej+1. The observation made in the beginning of the proof implies that e1 is equivalent to ej+1 in Pj∗Q and Pj+1∗Q. Thus, e1(Pj∗Q)e1≅ej+1(Pj∗Q)ej+1 and e1(Pj+1∗Q)e1≅ej+1(Pj+1∗Q)ej+1. Altogether, e1(Pj∗Q)e1≅Pej+1∗e1(Pj+1∗Q)e1. This implies that (4.2) holds for j+1 and, by induction, proves (4.2). For j=k, (4.2) gives the claim. □
To prove the proposition, assume first that I is finite. Take I={1,…,n}, for some n≥2. For 1≤i≤n, put \(B_{i}=\mathbb{C}p_{i,1}\oplus \cdots \oplus \mathbb{C}p_{i,k}\) and Ci=B1∗⋯∗Bi∗Ai+1∗⋯∗An. We claim that
The case i=1 follows from Claim 4.3. Assume that (4.3) holds for some 1≤i≤n−1. Since the projections pi,1 and pi+1,1 are equivalent in Ci by the observation made in the beginning of the proof of Claim 4.3, we get that pi,1Mpi,1≅pi+1,1Mpi+1,1 and pi,1Cipi,1≅pi+1,1Cipi+1,1. By applying Claim 4.2 to Ci=Ai+1∗(B1∗⋯∗Bi∗Ai+2∗⋯∗Ak) and the projections \((p_{i+1,j})_{j=1}^{k}\subset A_{i+1}\), we get that pi+1,1Cipi+1,1≅(∗1≤j≤kAi+1pi+1,j)∗pi+1,1Ci+1pi+1,1. The last three isomorphisms together imply that (4.3) holds for i+1. By induction, this proves (4.3).
Next, (4.3) for i=n gives that M1/k≅(∗1≤i≤n,1≤j≤kAipi,j)∗pn,1Cnpn,1. We will prove that
and thus finish the proof of case (1) by analyzing three separate cases.
If n=k=2, then \(C_{2}\cong L\mathbb{Z}\otimes \mathbb{M}_{2}(\mathbb{C})\) and [Dyk94, Proposition 3.2] impies that \(p_{2,1}C_{2}p_{2,1}\cong L\mathbb{Z}\). If n>2 or k>2, then \(C_{n}\cong L(*_{i=1}^{n}\mathbb{Z}/k\mathbb{Z})\) is a II1 factor. Since τ(pn,1)=1/k, we get that \(p_{n,1}C_{n}p_{n,1}\cong L(*_{i=1}^{n}\mathbb{Z}/k\mathbb{Z})^{1/k}\). Assume first that k=2 and n>2. Recall that \(L(*_{j=1}^{2}\mathbb{Z}/2\mathbb{Z})\cong L\mathbb{Z}\otimes \mathbb{M}_{2}( \mathbb{C})\) and \((A\otimes \mathbb{M}_{2}(\mathbb{C}))*L(\mathbb{Z}/2\mathbb{Z})\cong (A*L \mathbb{F}_{2})\otimes \mathbb{M}_{2}(\mathbb{C})\), for every tracial von Neumann algebra A, by [Dyk94, Theorem 3.5 (ii)]. Combining these facts with (4.1) and using induction gives that \(L(*_{i=1}^{n}\mathbb{Z}/2\mathbb{Z})\cong L\mathbb{F}_{n/2}\), thus \(L(*_{i=1}^{n}\mathbb{Z}/2\mathbb{Z})^{1/2}\cong L\mathbb{F}_{2n-3}\). Finally, assume that k>2. Then [Dyk93, Corollary 5.3] gives that \(L(\mathbb{Z}/k\mathbb{Z}*\mathbb{Z}/k\mathbb{Z})\cong L\mathbb{F}_{2(1-1/k)}\), while [Dyk93, Proposition 2.4] gives that \(L\mathbb{F}_{r}*L(\mathbb{Z}/k\mathbb{Z})\cong L\mathbb{F}_{r+1-1/k}\), for every r>1. By combining these facts, we get that \(L(*_{i=1}^{n}\mathbb{Z}/k\mathbb{Z})\cong L\mathbb{F}_{n(1-1/k)}\). Hence, using (4.1) we derive that \(L(*_{i=1}^{n}\mathbb{Z}/k\mathbb{Z})^{1/k}\cong L\mathbb{F}_{1+k^{2}[n(1-1/k)-1]}=L \mathbb{F}_{1+nk(k-1)-k^{2}}\). This altogether proves (4.4).
To treat case (2), assume that I is infinite. Take \(I=\mathbb{N}\). For i≥0, let Mi=A2i+1∗A2i+2. By applying case (1), we get that Mi is a II1 factor and \(M_{i}^{1/k}\cong (*_{i\leq l\leq i+1,1\leq j\leq k}A_{l}p_{l,j})*L \mathbb{F}_{(k-1)^{2}}\), for every i≥0. Since M=∗i≥0Mi, [DR00, Theorem 1.5] implies that \(M^{1/k}\cong *_{k\geq 0}M_{i}^{1/k}\). Thus, \(M^{1/k}\cong (*_{1\leq i,1\leq j\leq k}A_{i}p_{i,j})*L\mathbb{F}_{ \infty}\). Since ∗1≤i,1≤j≤kAipi,j is a free product of infinitely many II1 factors, it freely absorbs \(L\mathbb{F}_{\infty}\) by [DR00, Theorem 1.5]. This finishes the proof of case (2). □
We say that an abelian tracial von Neumann algebra (A,τ) is homogeneous if for every \(k\in \mathbb{N}\), there exists a partition of unity into k projections p1,…,pk∈A such that for every 1≤i≤k we have that τ(pi)=1/k and \((Ap_{i},k\;\tau _{|Ap_{i}})\) is isomorphic to (A,τ). A homogeneous abelian von Neumann algebra is necessarily diffuse. Also, note that \(L\mathbb{Z}\) and \((L\mathbb{Z})^{\omega}\) are homogenenous, and that the direct sum of two homogeneous abelian von Neumann algebras is homogeneous.
Corollary 4.4
Let A be a homogeneous abelian tracial von Neumann algebra. Then we have:
-
(1)
If 2≤n<∞ and k≥1, then \((A^{*n})^{1/k} \simeq A^{*nk} * L\mathbb{F}_{1+nk(k-1)-k^{2}}\).
-
(2)
\(\mathbb{Q}\subset \mathcal {F}(A^{*\infty})\).
Proof
Part (1) follows from Proposition 4.1. Proposition 4.1 also implies that \(1/k\in \mathcal {F}(A^{*\infty})\), for every \(k\in \mathbb{N}\), and thus part (2) also follows. □
When A is separable (and thus \(A\cong L\mathbb{Z}\)), Corollary 4.4 recovers two results of Voiculescu [Voi90]: the amplification formula \(L\mathbb{F}_{n}^{1/k}\cong L\mathbb{F}_{nk^{2}-k+1}\) and the fact that \(\mathbb{Q}\subset \mathcal {F}(L\mathbb{F}_{\infty})\). Corollary 4.4 extends these results to non-separable homogenenous abelian von Neumann algebras A. Recall that Radulescu [Rad92] showed that in fact \(\mathcal {F}(L\mathbb{F}_{\infty})=\mathbb{R}_{+}^{*}\). By analogy with this result, we expect that \(\mathcal {F}(A^{*\infty})=\mathbb{R}_{+}^{*}\), for any homogenenous abelian von Neumann algebras A.
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Funding
RB is supported by ANR grant AODynG 19-CE40-0008. DD holds the postdoctoral fellowship fundamental research 12T5221N of the Research Foundation Flanders. AI was supported by NSF DMS grants 1854074 and 2153805, and a Simons Fellowship. SP Supported in part by NSF Grant DMS-1955812 and the Takesaki Endowed Chair at UCLA.
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Boutonnet, R., Drimbe, D., Ioana, A. et al. Non-isomorphism of A∗n,2≤n≤∞, for a non-separable abelian von Neumann algebra A. Geom. Funct. Anal. 34, 393–408 (2024). https://doi.org/10.1007/s00039-024-00669-8
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DOI: https://doi.org/10.1007/s00039-024-00669-8