Non-isomorphism of A^∗n,2≤n≤∞, for a non-separable abelian von Neumann algebra A

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.

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∗⋯∗A_n, with A₁,A₂,…,A_n 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 p_i∈B such that Bp_i is unitarily conjugate to a direct summand of A_i, for every 1≤i≤n (see Corollary 3.7). This implies that the family {A_ip_i}_i, consisting of the maximal purely non-separable direct summands of A_i, 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 B_j of M. The uniqueness (up to unitary conjugacy, cutting and gluing) of this family ensures that the sans-rank of M, defined by

$$ \mathrm{r}_{ns}(M):=\sum _{j} \tau (1_{B_{j}})\in [0,+\infty ], $$

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 r_ns(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 r_ns(M^t)=r_ns(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 r_ns(∗_i∈IM_i)=∑_i∈Ir_ns(M_i), for any family M_i,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 E_A: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 ∥E_B(x^∗u_ny)∥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 p_i∈M is a projection, A_i⊂p_iMp_i is a singular MASA, for every i∈I, and A_i,A_i′ 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∈A_i, there exists j∈J such that A_ip≺_MB_j. 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∈B_j with B_jq⊀_MA_i, for every i∈I. As B_jq⊂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 A_i 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 r_ns(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 r_ns(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 r_ns(M) as a cardinality ≤|K|.

Remark 2.11

If M is a separable II₁ factor, then we clearly have r_ns(M)=0. If A⊂M is a singular MASA and ω is a free ultrafilter on $\mathbb{N}$, then A^ω⊂M^ω is a purely non-separable singular MASA, see [Pop83c, 5.3]. Moreover, disjoint MASAs in M give rise to disjoint ultrapower MASAs in M^ω. 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 M^ω, which has cardinality $\mathfrak {c}$, one may have singular MASAs that are not of this form.

The expression of r_ns(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 r_ns(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

$$ \mathrm {r}_{ns}(M^{t}) = \mathrm {r}_{ns}(M)/t. $$

In particular, if 0<r_ns(M)<∞, then M has trivial fundamental group, $\mathcal {F}(M)=\{1\}$.

Proof

It is enough to argue that r_ns(qMq)=r_ns(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 B_j⊂q_jMq_j, for some q_j≤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 m_ρ:M→M such that m_ρ(x₁x₂⋯x_n)=ρⁿx₁x₂⋯x_n, for every $n\in \mathbb{N}$ and $x_{i}\in M_{i_{j}}\ominus \mathbb{C}1$, where i_j∈{1,2}, for every 1≤j≤n, and i_j≠i_j+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↦∥m_ρ(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≺_MM_i, 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^∗(M_i)₁v. Since ∥m_ρ(x)−x∥₂≤|ρ−1|, for every x∈(M_i)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 ∥m_1−1/n(u_n)∥₂≤1/n. Let Q⊂P be the separable von Neumann subalgebra generated by {u_n}_n≥1. Let ρ∈(0,1). Then for every n≥1 such that ρ≤1−1/n we have that ∥m_ρ(u_n)∥₂≤∥m_1−1/n(u_n)∥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⋯ξ_n, where $n\in \mathbb{N}$, $\xi _{i}\in \mathcal {B}_{i_{j}}$, for some i_j∈{1,2}, for every 1≤j≤n, and i_j≠i_j+1, for every 1≤j≤n−1. Then $\mathcal {B}=\mathcal {B}_{0}\cup \{1\}$ is an orthonormal basis of L²(M).

Let {x_k}_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 N_i of M_i 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 (p_k)_k∈K⊂A and unitaries (u_k)_k∈K⊂M such that ∑_k∈Kp_k=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^∗⊂M_i.

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≺_MM_i. By Theorem 2.1, we can find nonzero projections e∈Aq,f∈M_i, a nonzero partial isometry v∈fMe and a ∗-homomorphism θ:Ae→fM_if 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)⊂fM_if is diffuse, by applying [IPP08, Theorem 1.1] (see also [Pop83b, Remarks 6.3.2)]) we get that vv^∗∈fM_if. Finally, let u∈M be any unitary such that ur=v. Then uAru^∗=vArv^∗=vAev^∗=θ(Ae)vv^∗⊂M_i, which finishes the proof. □

We continue by generalizing Corollary 3.5 to arbitrary tracial free products.

Corollary 3.6

Let (M_i,τ_i), i∈I, be a collection of tracial von Neumann algebras, and denote by M=∗_i∈IM_i their free product. Let A⊂pMp be a purely non-separable MASA. Then there exist projections (p_k)_k∈K⊂A and unitaries (u_k)_k∈K⊂M such that ∑_k∈Kp_k=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∈JM_j. 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={i_m}_m≥1.

Let {p_k}_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 u_k∈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∈Kp_k=p. Put r:=p−(∑_k∈Kp_k).

Assume by contradiction that r≠0. We claim that

$$ \text{$Ar\nprec _{M}*_{m\leq n}M_{i_{m}}$, for every $n\geq 1$.}$$

(3.1)

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 {p_k}_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

$$ \text{$Ar\prec _{M}^{\text{f}}*_{m>n}M_{i_{m}}$, for every $n\geq 1$.}$$

(3.2)

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 ∥η_n∥₂→∥r∥₂, ∥x⋅η_n∥₂≤∥x∥₂ and ∥a⋅η_n−η_n⋅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 ∥ζ_n∥₂→∥r∥₂, ∥a⋅ζ_n∥₂≤∥a∥₂ and ∥a⋅ζ_n−ζ_n⋅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⋅ζ_n,ζ_n⋅a〉=0, for every n. Using that ∥a⋅ζ_n−ζ_n⋅a∥₂→0, we get that ∥a⋅ζ_n∥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 ∥ζ_n∥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 M_i 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 M_i is abelian, for every i∈I. Then there exist projections (q_i)_i∈I⊂A and unitaries (v_i)_i∈I⊂M such that ∑_i∈Iq_i=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 (p_k)_k∈K⊂A and unitaries (u_k)_k∈K⊂M such that ∑_k∈Kp_k=p and for every k∈K, $u_{k}Ap_{k}u_{k}^{*}\subset M_{i_{k}}$, for some i_k∈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 i_k=i_k′. Since A⊂pMp is singular and p_kp_k′=0, there are no nonzero projections s∈Ap_k,s′∈Ap_k′ such that As and As′ are unitarily conjugated in M. This implies that r_kr_k′=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 M_ie_i 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∈IM_i is the union of the sans cores of M_i,i∈I. This allows us to deduce that the sans rank of M is the sum of the sans ranks of M_i,i∈I.

Theorem 3.8

Let (M_i,τ_i), i∈I, be a colection of tracial von Neumann algebras, and denote by M=∗_i∈IM_i their free product. Then r_ns(M)=∑_i∈Ir_ns(M_i). 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 M_i is naturally a sans family in M, for every i∈I.

Proof

We have two inequalities to prove.

Inequality 1. r_ns(M)≥∑_i∈Ir_ns(M_i).

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⊂pM_ip is a MASA, then A⊂pMp is a MASA.

• (2) If A⊂pM_ip is a singular diffuse von Neumann subalgebra, then A⊂pMp is singular.

• (3) If A⊂pM_ip, B⊂qM_iq are von Neumann subalgebras with A≺_MB, then $A \prec _{M_{i}} B$.

• (4) If A⊂pM_ip and B⊂qM_jq 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 M_i. We view every (not necessarily unital) subalgebra of M_i 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,

$$ \text{r}_{\text{ns}}(M)\geq \text{d}(\mathcal {A})=\sum _{i\in I}\text{d}( \mathcal {A}_{i})=\sum _{i\in I}\text{r}_{\text{ns}}(M_{i}). $$

Inequality 2. r_ns(M)≤∑_i∈Ir_ns(M_i).

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 A_l gives projections $(p_{k,l})_{k\in K_{l}}$ and unitaries $(u_{k,l})_{k\in K_{l}}$ such that for every k∈K_l 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 M_i of the form $u_{k,l}A_{l}p_{k,l}u_{k,l}^{*}$, for all l∈L,k∈K_l 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

$$ \text{r}_{\text{ns}}(M)=\text{d}(\mathcal {A})=\sum _{i\in I}\text{d}( \mathcal {A}_{i})\leq \sum _{i\in I} \text{r}_{\text{ns}}(M_{i}). $$

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 (A_i,τ_i), i∈I, be a collection of diffuse tracial abelian von Neumann algebras, and denote by M=∗_i∈IA_i their free product. For i∈I, let p_i∈A_i be the maximal (possibly zero) projection such that A_ip_i is purely non-separable. Then r_ns(M)=∑_i∈Iτ_i(p_i). Moreover, if |I|≥2 and ∑_i∈Iτ_i(p_i)∈(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 r_ns(A_i)=τ_i(p_i). 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, r_ns(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 s_k be a semicircular self-adjoint element belonging to A_k, the k^th copy of A in M. For an ℓ²-summable family of real numbers t=(t_k) with at least two non-zero entries, denote by A(t) the abelian von Neumann subalgebra of M generated by ∑_kt_ks_k. 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)⊀_MA_k, 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 A_k, 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 {B_i}_i are diffuse amenable von Neumann subalgebras of $L\mathbb{F}_{n}$ with B_i 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 p_i∈B_i and unitaries u_i∈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 A_k denotes the k^th 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 p_k∈A_k and unitaries u_k∈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 θ_α 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↦θ_α 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=a^k, 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 θ_g on $\mathbb{F}_{n}$ defined by θ_g(a_i)=a_i, if 1≤i≤n−1, and θ_g(a_n)=ga_ng⁻¹, where a₁,…,a_n are the free generators of $\mathbb{F}_{n}$. Similarly, if M=A₁∗⋯∗A_n, with A_i abelian diffuse, and n≥3, then any non-scalar unitary u∈A₁∗⋯∗A_n−1∗1 gives rise to an outer automorphism θ_u of M defined by θ_u(x)=x, if x∈A₁∗⋯∗A_n−1∗1, and θ_u(x)=uxu^∗, if x∈1∗A_n.

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 (A_i,τ_i), i∈I, be a countable collection of diffuse tracial abelian von Neumann algebras. Put M=∗_i∈IA_i and assume that |I|≥2. Let k≥2 and for every i∈I, let p_i,1,…,p_i,k∈A_i be projections such that τ(p_i,j)=1/k, for every 1≤j≤k, and $\sum _{j=1}^{k}p_{i,j}=1$.

Then M is a II₁ factor and M^1/k≅(∗_{i∈I,1≤j≤k}A_ip_i,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

$$ \begin{aligned} &\text{$L\mathbb{F}_{r}*L\mathbb{F}_{r'}\cong L\mathbb{F}_{r+r'}$; and}\\ &\text{$(L\mathbb{F}_{r})^{t}\cong L\mathbb{F}_{1+\frac{(r-1)}{t^{2}}}$, for every $1\leq r,r'\leq \infty $ and $t>0$.}\end{aligned} $$

(4.1)

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,…,e_k∈P and f₁,…,f_k∈Q such that e_i is central in P, τ(e_i)=τ(f_i)=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 e_j 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

$$ \text{$e_{1}(P*Q)e_{1}\cong Pe_{1}*\cdots *Pe_{j}*e_{1}(P_{j}*Q)e_{1}$, for every $1\leq j\leq k$}.$$

(4.2)

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 e_j+1∈P_j is a central projection, P_je_j+1=Pe_j+1 and $\mathbb{C}e_{j+1}\oplus P_{j}(1-e_{j+1})=P_{j+1}$, Fact 4.2 gives that e_j+1(P_j∗Q)e_j+1≅Pe_j+1∗e_j+1(P_j+1∗Q)e_j+1. The observation made in the beginning of the proof implies that e₁ is equivalent to e_j+1 in P_j∗Q and P_j+1∗Q. Thus, e₁(P_j∗Q)e₁≅e_j+1(P_j∗Q)e_j+1 and e₁(P_j+1∗Q)e₁≅e_j+1(P_j+1∗Q)e_j+1. Altogether, e₁(P_j∗Q)e₁≅Pe_j+1∗e₁(P_j+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 C_i=B₁∗⋯∗B_i∗A_i+1∗⋯∗A_n. We claim that

$$ \text{$p_{i,1}Mp_{i,1}\cong (*_{1\leq l\leq i,1\leq j\leq k}A_{l}p_{l,j})*p_{i,1}C_{i}p_{i,1}$, for every $1\leq i\leq n$.} $$

(4.3)

The case i=1 follows from Claim 4.3. Assume that (4.3) holds for some 1≤i≤n−1. Since the projections p_i,1 and p_i+1,1 are equivalent in C_i by the observation made in the beginning of the proof of Claim 4.3, we get that p_i,1Mp_i,1≅p_i+1,1Mp_i+1,1 and p_i,1C_ip_i,1≅p_i+1,1C_ip_i+1,1. By applying Claim 4.2 to C_i=A_i+1∗(B₁∗⋯∗B_i∗A_i+2∗⋯∗A_k) and the projections $(p_{i+1,j})_{j=1}^{k}\subset A_{i+1}$, we get that p_i+1,1C_ip_i+1,1≅(∗_1≤j≤kA_i+1p_i+1,j)∗p_i+1,1C_i+1p_i+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 M^1/k≅(∗_{1≤i≤n,1≤j≤k}A_ip_i,j)∗p_n,1C_np_n,1. We will prove that

$$ p_{n,1}C_{n}p_{n,1}\cong L\mathbb{F}_{nk(k-1)-k^{2}+1}$$

(4.4)

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(p_n,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 M_i=A_2i+1∗A_2i+2. By applying case (1), we get that M_i 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≥0M_i, [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≤k}A_ip_i,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,…,p_k∈A such that for every 1≤i≤k we have that τ(p_i)=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.