Partial Classification of the Baumslag-Solitar

Group Von Neumann Algebras

Niels Meesschaert^{1} and Stefaan Vaes^{2}

Received: March 28, 2013 Revised: March 12, 2014

Communicated by Joachim Cuntz

Abstract. We prove that the rational number |n/m| is an invariant of the group von Neumann algebra of the Baumslag-Solitar group BS(n, m). More precisely, if L(BS(n, m)) is isomorphic with L(BS(n′, m′)), then |n′/m′| = |n/m|^{±1}. We obtain this result by associating to abelian, but not maximal abelian, subalgebras of a II_{1} factor, an equivalence relation that can be of type III. In particular, we associate to L(BS(n, m)) a canonical equivalence relation of type III_{|n/m|}.

2010 Mathematics Subject Classification: Primary 46L36; Secondary 20E06, 22D25.

1. Introduction and statement of the main result

Some of the deepest open problems in functional analysis center around the classification of group von Neumann algebras L(G) associated with certain natural families of countable groups G. In the case of the free groups, this becomes the famous free group factor problem asking whether L(𝔽_{n}) ≅ L(𝔽_{m}) when n, m ≥ 2 and n ≠ m. For property (T) groups with infinite conjugacy classes (icc), this leads to Connes’s rigidity conjecture ([Co80]) asserting that an isomorphism L(G) ≅ L(Λ) between the property (T) factors entails an isomorphism G ≅ Λ of the groups. As a consequence of Connes’s uniqueness theorem of injective II_{1} factors ([Co75]), the group von Neumann algebra L(G) of an amenable icc group G is isomorphic with the unique hyperfinite II_{1} factor R. In the nonamenable case, many nonisomorphic groups G are known to have nonisomorphic group von Neumann algebras L(G). Nevertheless, concerning the classification of group von Neumann algebras of natural families of groups, e.g. lattices in simple Lie groups, little is known. A notable exception however is [CH88] where it is shown that for n ≠ m, lattices in Sp(n, 1), respectively Sp(m, 1), have nonisomorphic group von Neumann algebras. Since 2001, Popa has been developing a new arsenal of techniques to study II_{1} factors, called deformation/rigidity theory. This theory has provided several classes G of groups such that an isomorphism L(G) ≅ L(Λ) with both G, Λ ∈ G entails the isomorphism G ≅ Λ. By [Po04], this holds in particular when G is the class of wreath product groups of the form (ℤ/2ℤ)≀Γ with Γ an icc property (T) group. In [IPV10], the first W^{∗}-superrigidity theorems for group von Neumann algebras were discovered, yielding icc groups G such that an isomorphism L(G) ≅ L(Λ) with Λ an arbitrary countable group, implies that G ≅ Λ. The groups G discovered in [IPV10] are generalized wreath products of a special form. In [BV12], it was then shown that one can actually take G = (ℤ/2ℤ)^{(Γ)} ⋊ (Γ × Γ) with Γ ranging over a large family of nonamenable groups including the free groups 𝔽_{n}, n ≥ 2. In this article, we apply Popa’s deformation/rigidity theory to partially classify the group von Neumann algebras of the Baumslag-Solitar groups BS(n, m). Recall that for all n, m ∈ ℤ − {0}, this group is defined as the group generated by a and b subject to the relation ba^{n}b^{−1} = a^{m}. So,

1Supported by ERC Starting Grant VNALG-200749 and Research Programme G.0639.11 of the Research Foundation –Flanders (FWO).

2Supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/13/079.

Documenta Mathematica 19 (2014) 629–645

BS(n, m) := ⟨a, b | ba^{n}b^{−1} = a^{m}⟩ .

The Baumslag-Solitar groups were introduced in [BS62] as the first examples of finitely presented non-Hopfian groups. Ever since, they have been used as examples and counterexamples for numerous group theoretic phenomena. Therefore, it is a natural problem to classify the group von Neumann algebras L(BS(n, m)). Whenever |n| = 1 or |m| = 1, the group BS(n, m) is solvable, hence amenable. So we always assume that |n| ≥ 2 and |m| ≥ 2. In that case, BS(n, m) contains a copy of the free group 𝔽_{2} and hence, is nonamenable. In [Mo91], the Baumslag-Solitar groups were classified up to isomorphism: BS(n, m) ≅ BS(n′, m′) if and only if {n, m} = {εn′, εm′} for some ε ∈ {−1, 1}. So, up to isomorphism, we only consider 2 ≤ n ≤ |m|. Finally by [St05, Exemple 2.4], the group BS(n, m) is icc if and only if |n| ≠ |m|. Therefore, we always assume that 2 ≤ n < |m|. Using Popa’s deformation/rigidity theory and in particular his spectral gap rigidity ([Po06]) and the work on amalgamated free products ([IPP05]), several structural properties of the II_{1} factors M = L(BS(n, m)) were proven. In particular, it was shown in [Fi10] that M is not solid, that M is prime and that M has no Cartan subalgebra. More generally, it is proven in [Fi10] that any amenable regular von Neumann subalgebra of M must have a nonamenable relative commutant.

Our main result is the following partial classification theorem for the Baumslag-Solitar group von Neumann algebras L(BS(n, m)). Whenever M is a II_{1} factor and t > 0, we denote by M^{t} the amplification of M. Up to unitary conjugacy, M^{t} is defined as p(M_{n}(ℂ)⊗M)p where p is a projection satisfying (Tr ⊗τ)(p) = t. The II_{1} factors M and N are called stably isomorphic if there exists a t > 0 such that M ≅ N^{t}.

Theorem A. Let n, m, n′, m′ ∈ ℤ such that 2 ≤ n < |m| and 2 ≤ n′ < |m′|. If L(BS(n, m)) is stably isomorphic with L(BS(n′, m′)), then \frac{n}{|m|} = \frac{n′}{|m′|}.

Note that Theorem A formally resembles, but is independent of, the results in [Ki11] on orbit equivalence relations of essentially free ergodic probability measure preserving actions of Baumslag-Solitar groups, especially [Ki11, Proposition B.2 and Theorem 1.2]. It would be very interesting to find a framework that unifies both types of results. We prove our Theorem A by associating a canonical equivalence relation to L(BS(n, m)) and proving that it is of type III_{n/|m|}. More precisely, assume that (M, τ) is a von Neumann algebra with separable predual, equipped with a faithful normal tracial state. Whenever A ⊂ M is an abelian von Neumann subalgebra, the normalizer

N_{M}(A) := {u ∈ U(M) | uAu^{∗} = A}

induces a group of trace preserving automorphisms of A. Writing A = L^{∞}(X, μ) with μ being induced by τ_{|A}, the corresponding orbit equivalence relation is a countable probability measure preserving (pmp) equivalence relation on (X, μ). More generally, we can consider the set of partial isometries

{u ∈ M | u^{∗}u and uu^{∗} are projections in A′ ∩ M and uAu^{∗} = Auu^{∗} } . (1)

Every such partial isometry induces a partial automorphism of A and hence a partial automorphism of (X, μ). We denote by R(A ⊂ M) the equivalence relation generated by all these partial automorphisms. When A ⊂ M is maximal abelian, i.e. A′ ∩ M = A, then R(A ⊂ M) coincides with the orbit equivalence relation induced by the normalizer N_{M}(A). In particular, in that case the equivalence relation R(A ⊂ M) preserves the probability measure μ. If however A ⊂ M is not maximal abelian, the partial automorphisms of A induced by the partial isometries in the set (1) need not be trace preserving. So in general, R(A ⊂ M) can be an equivalence relation of type III. Our main technical result is Theorem 3.3 below, roughly saying the following. If A, B ⊂ M are abelian subalgebras such that Z(A′ ∩ M) = A and Z(B′ ∩ M) = B, and if there exist intertwining bimodules A ≺ B and B ≺ A (in the sense of Popa, see [Po03] and Theorem 2.3 below), then the equivalence relations R(A ⊂ M) and R(B ⊂ M) must be stably isomorphic. In particular, their types must be the same.

In Section 4, we apply this to M = L(BS(n, m)) and A equal to the abelian von Neumann subalgebra generated by the unitary u_{a}. We prove that R(A ⊂ M) is the unique hyperfinite ergodic equivalence relation of type III_{n/|m|}. The proof of Theorem A can then be outlined as follows. First we note that the von Neumann algebra A′ ∩ M is nonamenable. Conversely if Q ⊂ M is a nonamenable subalgebra, it was proven in [CH08], using spectral gap rigidity ([Po06]) and the structure theory of amalgamated free product factors ([IPP05]), that Q′ ∩ M ≺ A. So, up to intertwining-by-bimodules, the position of A inside M is “canonical”. Therefore a stable isomorphism between L(BS(n, m)) and L(BS(n′, m′)) will preserve, up to intertwining-by-bimodules, these canonical abelian subalgebras. Hence their associated equivalence relations are stably isomorphic and, in particular, have the same type. This gives us the equality n/|m| = n′/|m′|.

2. Preliminaries

We denote by (M, τ) a von Neumann algebra equipped with a faithful normal tracial state τ. We always assume that M has a separable predual. If B is a von Neumann subalgebra of (M, τ), we denote by E_{B} the unique trace preserving conditional expectation of M onto B. Whenever x ∈ M is a normal element, we denote by supp(x) its support, i.e. the smallest projection p ∈ M that satisfies xp = x (or equivalently, px = x). Let R be a countable nonsingular (i.e. measure class preserving) equivalence relation on a standard probability space (X, μ). We denote by [[R]] the full pseudogroup of R, i.e. the pseudogroup of all partial nonsingular automorphisms φ of X such that the graph of φ is contained in R. We denote the domain of φ by dom(φ) and its range by ran(φ). We denote by [x] the equivalence class of x ∈ X. Assume that also R′ is a countable nonsingular equivalence relation on the standard probability space (X′, μ′). The equivalence relations R and R′ are called

• isomorphic, if there exists a nonsingular isomorphism Δ : X → X′ such that Δ([x]) = [Δ(x)] for almost every x ∈ X ;

• stably isomorphic, if there exist Borel subsets Z ⊂ X and Z′ ⊂ X′ that meet almost every orbit and a nonsingular isomorphism Δ : Z → Z′ such that Δ([x] ∩ Z) = [Δ(x)] ∩ Z′ for almost every x ∈ Z.

2.1. HNN extensions and Baumslag-Solitar groups Let G be a group, H < G a subgroup and θ : H → G an injective group homomorphism. The HNN extension HNN(G, H, θ) is defined as the group generated by G and an additional element t subject to the relation θ(h) = tht^{−1} for all h ∈ H. So,

HNN(G, H, θ) = ⟨G, t | θ(h) = tht^{−1} for all h ∈ H⟩ .

Elements of HNN(G, H, θ) can be canonically written as “reduced words” using as letters the elements of G and the letters t^{±1}. More precisely, we have the following lemma.

Lemma 2.1 (Britton’s lemma, [Br63]). Consider the expression g = g_{0}t^{n_{1}}g_{1}t^{n_{2}} ⋯ t^{n_{k}}g_{k} with k ≥ 0, g_{0}, g_{k} ∈ G, g_{1}, …, g_{k−1} ∈ G − {e} and n_{1}, …, n_{k} ∈ ℤ − {0}. We call this expression reduced if the following two conditions hold:

• for every i ∈ {1, …, k − 1} with n_{i} > 0 and n_{i+1} < 0, we have g_{i} ∉ H,

• for every i ∈ {1, …, k − 1} with n_{i} < 0 and n_{i+1} > 0, we have g_{i} ∉ θ(H).

If the above expression for g is reduced, then g ≠ e in the group HNN(G, H, θ), unless k = 0 and g_{0} = e. In particular, the natural homomorphism of G to HNN(G, H, θ) is injective.

Recall from the introduction that the Baumslag-Solitar group BS(n, m) is defined for all n, m ∈ ℤ − {0} as

BS(n, m) := ⟨a, b | ba^{n}b^{−1} = a^{m}⟩ .

It is one of the easiest examples of an HNN extension. We also recall from the introduction that the BS(n, m) with 2 ≤ n < |m| form a complete list of all nonamenable icc Baumslag-Solitar groups up to isomorphism. Since we only want to consider the case where L(BS(n, m)) is a nonamenable II_{1} factor, we always assume that 2 ≤ n < |m|.

2.2. Hilbert bimodules and intertwining-by-bimodules If M and N are tracial von Neumann algebras, then a left M-module is a Hilbert space H endowed with a normal ∗-homomorphism π : M → B(H). A right N-module is a left N^{op}-module. An M-N-bimodule is a Hilbert space H endowed with commuting normal ∗-homomorphisms π : M → B(H) and φ : N^{op} → B(H). For x ∈ M, y ∈ N and ξ ∈ H, we write xξy instead of π(x)φ(y^{op})(ξ). We denote an M-N-bimodule H by _{M}H_{N}. We call an M-N-bimodule bifinite if it is finitely generated both as a left Hilbert M-module and a right Hilbert N-module. Let A and B be abelian von Neumann algebras. We denote by PIso(A, B) the set of all partial isomorphisms from A to B, i.e. isomorphisms α : Aq → Bp, where q ∈ A and p ∈ B are projections. We write PAut(A) instead of PIso(A, A). Note that to every α ∈ PIso(A, B) we can associate an A-B-bimodule _{A}H(α)_{B} given by H(α) = L^{2}(Bp) and aξb = α(aq) ξ bp. The composition of two partial isomorphisms is defined as follows: if α ∈ PIso(B, C) and β ∈ PIso(A, B) are given by α : Bp → Cr and β : Aq → Bp′ for projections q ∈ A, p, p′ ∈ B and r ∈ C, then the composition α ∘ β ∈ PIso(A, C) is defined by x ↦ α(β(x)) for all x ∈ Aqβ^{−1}(pp′). The following is a well known result.

Lemma 2.2. Let A and B be abelian von Neumann algebras. Then every bifinite A-B-bimodule _{A}H_{B} is isomorphic to a direct sum of bimodules of the form AH(α)_{B} with α ∈ PIso(A, B).

We finally recall Popa’s intertwining-by-bimodules theorem.

Theorem 2.3 ([Po03, Theorem 2.1 and Corollary 2.3]). Let (M, τ) be a tracial von Neumann algebra and let A, B ⊂ M be possibly nonunital von Neumann subalgebras. Denote their respective units by 1_{A} and 1_{B}. The following three conditions are equivalent:

1. 1_{A}L^{2}(M)1_{B} admits a nonzero A-B-subbimodule that is finitely generated as a right B-module.

2. There exist nonzero projections p ∈ A, q ∈ B, a normal unital ∗- homomorphism ψ : pAp → qBq and a nonzero partial isometry v ∈ pMq such that av = vψ(a) for all a ∈ pAp.

3. There is no sequence of unitaries u_{n} ∈ U(A) satisfying ||E_{B}(xu_{n}y^{∗})||_{2} → 0 for all x, y ∈ 1_{B}M1_{A}.

If one of these equivalent conditions holds, we write A ≺_{M} B.

2.3. Quasi-regularity Let (M, τ) be a tracial von Neumann algebra and N ⊂ M a von Neumann subalgebra. We denote by QN_{M}(N) the quasi-normalizer of N inside M, i.e. the unital ∗-algebra defined by

{a ∈ M | ∃b_{1}, …, b_{k} ∈ M, ∃d_{1}, …, d_{r} ∈ M

such that Na ⊂ ∑_{i=1}^{k} b_{i}N and aN ⊂ ∑_{j=1}^{r} Nd_{j}} .

We call N ⊂ M quasi-regular if QN_{M}(N)′′ = M. If A, B ⊂ M are abelian von Neumann subalgebras, we define Q_{M}(A, B) as

Q_{M}(A, B) := {v ∈ M | vv^{∗} ∈ A′ ∩ M , v^{∗}v ∈ B′ ∩ M and Av = vB} .

Whenever v ∈ Q_{M}(A, B), we define q_{v} = supp(E_{A}(vv^{∗})) and p_{v} = supp(E_{B}(v^{∗}v)), and we denote by α_{v} : Aq_{v} → Bp_{v} the unique ∗-isomorphism satisfying av = vα_{v}(a) for all a ∈ Aq_{v}. Note that the set Q_{M}(A, B) can be {0}. In Lemma 2.4, we will see that Q_{M}(A, B) ≠ {0} if and only if there exists a bifinite A-B-subbimodule _{A}H_{B} of AL^{2}(M)_{B}. We denote Q_{M}(A, A) by Q_{M}(A).

Lemma 2.4. Let (M, τ) be a tracial von Neumann algebra and A, B ⊂ M abelian von Neumann subalgebras. Then the following statements hold.

1. If α ∈ PIso(A, B) and if θ : _{A}H(α)_{B} → _{A}L^{2}(M)_{B} is an A-B-bimodular isometry, then there exists a partial isometry v ∈ Q_{M}(A, B) such that α = α_{v} and such that

θ(H(α)) ⊂ \overline{v(B′ ∩ M)}^{||·||_{2}} ⊂ \overline{span}^{||·||_{2}} Q_{M}(A, B) .

2. Every bifinite A-B-subbimodule _{A}H_{B} of _{A}L^{2}(M)_{B} is contained in \overline{span}^{||·||_{2}} Q_{M}(A, B).

3. Q_{M}(A)′′ = QN_{M}(A)′′.

4. We have Q_{M}(A, B) ≠ {0} if and only if _{A}L^{2}(M)_{B} admits a nonzero bifinite A-B-subbimodule.

Proof. 1. Let α : Aq → Bp be an element of PIso(A, B). Define ξ := θ(p) ∈ L^{2}(M) and let ξ = v|ξ| be its polar decomposition. For all a ∈ A, we have aξ = ξα(a) and hence, av = vα(a). Furthermore p = supp(E_{B}(v^{∗}v)) and q = α^{−1}(p) = supp(E_{A}(vv^{∗})). So we find that v ∈ Q_{M}(A, B) and α = α_{v}.

Because |ξ| ∈ L^{2}(B′∩M), we have that ξ = v|ξ| is an element of \overline{v(B′ ∩ M)}^{||·||_{2}}. Since p generates _{A}H(α)_{B} as a right Hilbert B-module, we have proven the first

inclusion θ(H(α)) ⊂ \overline{v(B′ ∩ M)}^{||·||_{2}}. Since v ∈ Q_{M}(A, B), also v(B′ ∩ M) ⊂ Q_{M}(A, B) and the second inclusion in statement 1 is proven as well.

2. Let _{A}H_{B} be a bifinite A-B-subbimodule of _{A}L^{2}(M)_{B}. By Lemma 2.2, AH_{B} is isomorphic to a direct sum of bimodules of the form _{A}H(α_{i})_{B} with α_{i} ∈ PIso(A, B). Using statement 1 of the lemma, we find that H is generated by subspaces of \overline{span}^{||·||_{2}} Q_{M}(A, B). This proves statement 2.

3. By definition, we have Q_{M}(A)′′ ⊂ QN_{M}(A)′′. On the other hand, AL^{2}(QN_{M}(A)′′)_{A} is a direct sum of bifinite A-A-subbimodules of _{A}L^{2}(M)_{A}. So by statement 2, we have that L^{2}(QN_{M}(A)′′) ⊂ \overline{span}^{||·||_{2}}(Q_{M}(A)). Therefore we conclude that QN_{M}(A)′′ = Q_{M}(A)′′. Finally, 4 is an immediate consequence of 2.

We end this subsection with the following lemma, clarifying why later, we will consider abelian subalgebras A ⊂ M satisfying Z(A′ ∩ M) = A. Note that since A is abelian, the condition Z(A′ ∩ M) = A is equivalent with the “bicommutant” property (A′ ∩ M)′ ∩ M = A. Also note that the composition of two partial isomorphisms was defined before Lemma 2.2.

Lemma 2.5. Let (M, τ) be a tracial von Neumann algebra and A, B, C ⊂ M abelian von Neumann subalgebras. If v ∈ Q_{M}(A, B), w ∈ Q_{M}(B, C) and if Z(B′ ∩ M) = B, then there exists an element u ∈ Q_{M}(A, C) such that α_{w} ∘ α_{v} = α_{u}.

Proof. Choose v ∈ Q_{M}(A, B) and w ∈ Q_{M}(B, C). Note that vbw ∈ Q_{M}(A, C) for every b ∈ B′ ∩M and α_{vbw} = α_{w} ∘α_{v}_{|Aq_{vbw}}. We claim that ⋁_{b∈B′∩M} q_{vbw} = α_{v}^{−1}(q_{w}p_{v}).

Denote α_{v}^{−1}(q_{w}p_{v}) − ⋁_{b∈B′∩M} q_{vbw} by r. We need to prove that r is zero. Since (B′ ∩ M)′ ∩ M = B, we have that for every x ∈ M, the projection supp(E_{B}(xx^{∗})) equals the projection of L^{2}(M) onto the closed linear span of (B′ ∩ M)xM ⊂ L^{2}(M). Since w^{∗}bv^{∗}r = 0 for every b ∈ B′ ∩ M, it follows that w^{∗}q_{v^{∗}r} = 0. Therefore q_{w} is orthogonal to q_{v^{∗}r}. Because q_{v^{∗}r} = α_{v}(r) and α_{v}(r) ≤ q_{w}, it follows that α_{v}(r) = 0. Hence r = 0 and our claim that ⋁_{b∈B′∩M} q_{vbw} = α_{v}^{−1}(q_{w}p_{v}) is proven. By cutting down with appropriate projections, we find b_{n} ∈ B′ ∩ M such that the projections q_{vb_{n}w} are orthogonal and sum up to α_{v}^{−1}(q_{w}p_{v}). In particular, the left supports, resp. right supports, of the elements vb_{n}w are orthogonal. So we can define u = ∑_{n} vb_{n}w. It follows that u ∈ Q_{M}(A, C) and α_{w} ∘ α_{v} = α_{u}.

2.4. The type of an ergodic nonsingular countable equivalence

relation Let R be a nonsingular ergodic countable Borel equivalence relation on a standard probability space (X, μ). Using the map π : R → X : π(x, y) = x, we define the measure μ^{(1)} on R given by

μ^{(1)}(U) = ∫_{X} #(U ∩ π^{−1}(x)) dμ(x) for all Borel sets U ⊂ R .

We define R^{(2)} := {(x, y, z) ∈ X^{3} | (x, y), (y, z) ∈ R}. Similarly, using the map ρ : R^{(2)} → X : ρ(x, y, z) = x, we define the measure μ^{(2)} on R^{(2)} given by

μ^{(2)}(V) = ∫_{X} #(V ∩ ρ^{−1}(x)) dμ(x) for all Borel sets V ⊂ R^{(2)} .

The Radon-Nikodym 1-cocycle of R is the μ^{(1)}-a.e. uniquely defined Borel map ω : R → ℝ such that

ω(φ(x), x) = log(\frac{dμ ∘ φ}{dμ}(x)) for all φ ∈ [[R]] and almost every x ∈ dom φ .

Note that ω satisfies the 1-cocycle relation ω(x, z) = ω(x, y) + ω(y, z) for μ^{(2)}- a.e. (x, y, z) ∈ R^{(2)}. One then defines the Maharam extension R̃ of R as the equivalence relation on (X × ℝ, μ × exp(−t)dt) defined by

(x, t) ∼ (y, s) if and only if (x, y) ∈ R and t − s = ω(x, y).

Note that μ × exp(−t)dt is an infinite invariant measure for R̃. Denote the von Neumann algebra of all R̃-invariant functions in L^{∞}(X × ℝ) by L^{∞}(X × ℝ)R̃. Since R was assumed to be ergodic, one can easily check that the action of ℝ on L^{∞}(X × ℝ)R̃ given by translation of the second variable, is also ergodic. Depending on how this action of ℝ looks like, we define as follows the type of R.

• I or II, if the action is conjugate with ℝ ↷ ℝ ; • III_{λ} (0 < λ < 1), if the action is conjugate with ℝ ↷ ℝ/ℤ log(λ) ;

• III_{1}, if the action is on one point ;

• III_{0}, if the action is properly ergodic, i.e. is ergodic and has orbits of measure zero.

Remark 2.6. Denote by L(R) the von Neumann algebra associated with R. Denote by φ the normal semifinite faithful state on L(R) that is induced by μ. Finally denote by (σ_{t}^{φ})_{t∈ℝ} its modular automorphism group. There is a canonical identification L(R̃) ≅ L(R) ⋊_{σ^{φ}} ℝ. Under this identification, the dual action of ℝ on L(R) ⋊_{σ^{φ}} ℝ corresponds to the action of ℝ on L(R̃) that we defined above. Also, the center of L(R)⋊_{σ^{φ}} ℝ corresponds to L^{∞}(X ×ℝ)R̃. Altogether it follows that the type of the equivalence relation R coincides with the type of the factor L(R).

Lemma 2.7. Let R be a nonsingular ergodic countable Borel equivalence relation on the standard probability space (X, μ). Denote by ω its Radon-Nikodym 1cocycle. If the essential image Im(ω) of ω equals log(λ)ℤ for some 0 < λ < 1 and if the kernel Ker(ω) of ω is an ergodic equivalence relation, then R is of type III_{λ}.

Proof. Since Ker(ω) is an ergodic equivalence relation on (X, μ), we have

L^{∞}(X × ℝ)R̃ ⊂ L^{∞}(X × ℝ)^{Ker(ω)} = 1 ⊗ L^{∞}(ℝ) .

For a given F ∈ L^{∞}(ℝ), we have that 1 ⊗ F is R̃-invariant if and only if F is invariant under translation by the essential image of ω. So,

L^{∞}(X × ℝ)R̃ = 1 ⊗ L^{∞}(ℝ/ log(λ)ℤ) .

3. Equivalence relations associated to subalgebras that are abelian, but not maximal abelian

Throughout this section, we fix a tracial von Neumann algebra (M, τ) with separable predual. We also fix an abelian von Neumann subalgebra A ⊂ M satisfying Z(A′ ∩ M) = A. Choose a standard probability space (X, μ) such that A = L^{∞}(X, μ). For every nonsingular partial automorphism φ of (X, μ), we denote by α_{φ} the corresponding partial automorphism of A. We first prove that Q_{M}(A) induces a nonsingular countable Borel equivalence relation R(A ⊂ M) on (X, μ). For this, we introduce the notation

G(A ⊂ M) := {α_{v} | v ∈ Q_{M}(A)} .  (2)

Proposition 3.1. There exists a nonsingular countable Borel equivalence relation R on (X, μ) with the following property: a nonsingular partial automorphism φ of X satisfies α_{φ} ∈ G(A ⊂ M) if and only if (x, φ(x)) ∈ R for a.e. x ∈ dom(φ).

Moreover, R is essentially unique: if a nonsingular countable Borel equivalence relation R′ on (X, μ) satisfies the same property, then there exists a Borel subset X_{0} ⊂ X with μ(X − X_{0}) = 0 and R_{|X_{0}} = R′_{|X_{0}}. We denote R(A ⊂ M) := R. The equivalence relation R(A ⊂ M) is ergodic if and only if QN_{M}(A)′′ is a factor.

Before proving Proposition 3.1, we introduce some terminology and a lemma. To every α ∈ PAut(A) are associated the support projections q_{α}, p_{α} ∈ A such that α : Aq_{α} → Ap_{α} is a ∗-isomorphism. Assume that α ∈ PAut(A) and F ⊂ PAut(A). We say that α is a gluing of elements in F, if there exists a sequence of elements α_{n} ∈ F and projections q_{n} ∈ A such that q_{α} = ∑_{n} q_{n} and such that q_{n} ≤ q_{α_{n}} and α_{|Aq_{n}} = α_{n}_{|Aq_{n}} for all n.

Lemma 3.2. Let J ⊂ Q_{M}(A) and v ∈ Q_{M}(A) such that v ∈ \overline{span}^{||·||_{2}}J. Then α_{v} is a gluing of elements in {α_{w} | w ∈ J }.

Proof. By a standard maximality argument, it suffices to prove that for every nonzero projection q ∈ Aq_{v}, there exists a nonzero subprojection q_{0} ∈ Aq and a w ∈ J such that q_{0} ≤ q_{w} and α_{v}_{|Aq_{0}} = α_{w}_{|Aq_{0}}. So fix a nonzero projection q ∈ Aq_{v}. It follows that qE_{A}(vv^{∗}) ≠ 0. Since v ∈ \overline{span}^{||·||_{2}}J, we can pick a w ∈ J such that qE_{A}(vw^{∗}) ≠ 0. Define q_{1} := supp(E_{A}(vw^{∗})) and note that q_{1} ∈ q_{v}Aq_{w} = Aq_{v}q_{w}. Also note that qq_{1} ≠ 0. For all a ∈ A, we have

α_{v}^{−1}(ap_{v}) vw^{∗} = v a w^{∗} = vw^{∗} α_{w}^{−1}(ap_{w}) .

Applying the conditional expectation onto A and using that A is abelian, we find that

α_{v}^{−1}(ap_{v}) q_{1} = α_{w}^{−1}(ap_{w}) q_{1} for all a ∈ A .

This means that α_{v}_{|Aq_{1}} = α_{w}_{|Aq_{1}}. We put q_{0} := qq_{1}. We already showed that q_{0} ≠ 0. Since q_{0} ≤ q_{1}, we have that α_{v}_{|Aq_{0}} = α_{w}_{|Aq_{0}}.

Proof of Proposition 3.1. We say that a subpseudogroup G ⊂ PAut(A) is of countable type if there exists a countable subset J ⊂ G such that every α ∈ G is a gluing of elements in J. To prove the first part of the proposition, we must show that G(A ⊂ M) is a subpseudogroup of countable type of PAut(A). From Lemma 2.5, it follows that G(A ⊂ M) is indeed a subpseudogroup. Since M has a separable predual, we can choose a countable ∥·∥_{2}-dense subset J ⊂ Q_{M}(A). By Lemma 3.2, every α ∈ G(A ⊂ M) is a gluing of elements in {α_{w} | w ∈ J }. Hence G(A ⊂ M) is of countable type. So the first part of the proposition is proven and we can essentially uniquely define the nonsingular countable Borel equivalence relation R on (X, μ). Since A′ ∩ M ⊂ QN_{M}(A)′′ and since we assumed that (A′ ∩ M)′ ∩ M = A, the center of QN_{M}(A)′′ is a subalgebra of A. By Lemma 2.4.3, we have QN_{M}(A)′′ = Q_{M}(A)′′. Therefore,

Z(QN_{M}(A)′′) = {a ∈ A | av = va for all v ∈ Q_{M}(A)} .

The right hand side equals A^{R}, the subalgebra of R-invariant functions in A. So R is ergodic if and only if QN_{M}(A)′′ is a factor.

For our application, the following theorem is crucial. It says that R(A ⊂ M) remains the same, up to stable isomorphism, if we replace A by an abelian subalgebra B that has a mutual intertwining bimodule into A.

Theorem 3.3. Let M be a II_{1} factor with separable predual. Let A, B ⊂ M be abelian, quasi-regular von Neumann subalgebras satisfying Z(A′ ∩ M) = A and Z(B′ ∩ M) = B. If A ≺_{M} B and B ≺_{M} A, then the equivalence relations R(A ⊂ M) and R(B ⊂ M) are stably isomorphic.

Proof. Since A, B are quasi-regular and since A ≺_{M} B as well as B ≺_{M} A, there exists a nonzero bifinite A-B-subbimodule of L^{2}(M). So by Lemma 2.4.4, there exists a nonzero element v ∈ Q_{M}(A, B) with corresponding α_{v} ∈ PAut(A, B). Using the notation in (2) and using Lemma 2.5, we find that

α_{v} ∘ β ∘ α_{v}^{−1} ∈ G(B ⊂ M) for all β ∈ G(A ⊂ M) and
α_{v}^{−1} ∘ γ ∘ α_{v} ∈ G(A ⊂ M) for all γ ∈ G(B ⊂ M) .

So α_{v} implements a stable isomorphism between R(A ⊂ M) and R(B ⊂ M).

The following lemma will allow us to easily compute R(A ⊂ M) in concrete examples.

Lemma 3.4. Let (M, τ) be a tracial von Neumann algebra and A ⊂ M an abelian von Neumann subalgebra satisfying Z(A′ ∩ M) = A. Let F ⊂ M be a subset such that

• M = (F ∪ F^{∗} ∪ (A′ ∩ M))′′,

• as an A-A-bimodule, \overline{span}^{||·||_{2}}AFA is isomorphic to a direct sum of bimodules of the form _{A}H(α_{n})_{A} with α_{n} ∈ PAut(A).

Choose nonsingular partial automorphisms φ_{n} of (X, μ) such that α_{n} = α_{φ_{n}} for all n. Up to measure zero, R(A ⊂ M) is generated by the graphs of the partial automorphisms φ_{n}.

Proof. We again use the notation (2). By Lemma 2.4.1, we find v_{n} ∈ Q_{M}(A) such that α_{n} = α_{v_{n}} and

\overline{span}^{||·||_{2}}AFA ⊂ \overline{span}^{||·||_{2}}\{v_{n}(A′ ∩ M) | n ∈ ℕ\} .  (3)

In particular, we have α_{n} ∈ G(A ⊂ M). Choose nonsingular partial automorphisms φ_{n} of (X, μ) such that α_{n} = α_{φ_{n}} for all n. Denote by R the smallest (up to measure zero) equivalence relation on (X, μ) that contains the graphs of all the partial automorphisms φ_{n}. By the previous paragraph, we know that R is a subequivalence relation of R(A ⊂ M). Denote by J the set of all products of elements in

{v_{n} | n ∈ ℕ} ∪ {v_{n}^{∗} | n ∈ ℕ} ∪ (A′ ∩ M) .

By construction, the graph of every α_{w}, w ∈ J, belongs to R. Combining our assumption that M = (F ∪ F^{∗} ∪ (A′ ∩ M))′′ with (3), it follows that span J is ∥ · ∥_{2}-dense in L^{2}(M). By Lemma 3.2, every α ∈ G(A ⊂ M) is a gluing of elements in {α_{w} | w ∈ J }. So the graph of every α ∈ G(A ⊂ M) belongs to R a.e. Hence R equals R(A ⊂ M) almost everywhere.

We finally note in the following proposition that every nonsingular countable Borel equivalence relation R arises as R(A ⊂ M).

Proposition 3.5. Let R be a nonsingular countable Borel equivalence relation. Then there exists a quasi-regular inclusion of an abelian von Neumann algebra A in a tracial von Neumann algebra (M, τ) satisfying Z(A′ ∩M) = A and such that R ≅ R(A ⊂ M).

Proof. Let R be a nonsingular countable Borel equivalence relation on a standard probability space (X, μ). Denote by (P, Tr) the unique hyperfinite II_{∞} factor and choose a trace-scaling action (α_{t})_{t∈ℝ} of ℝ on P. This means that Tr ∘α_{t} = e^{−t} Tr. The corresponding action of ℝ on L^{2}(P) will also be denoted by (α_{t}). We denote by ω : R → ℝ the Radon-Nikodym 1-cocycle of R (see Section 2.4). In the same way as with the Maharam extension of a nonsingular group action, the equivalence relation R admits a natural trace preserving action on L^{∞}(X)\overline{⊗} P. We denote by (M, Tr) the crossed product. For completeness, we recall the construction of (M, Tr). To every φ ∈ [[R]], we associate the operator W_{φ} on L^{2}(R, L^{2}(P)) given by

(W_{φ}ξ)(x, y) = \begin{cases}α_{ω(x,φ^{−1}(x))}(ξ(φ^{−1}(x), y)) & if x ∈ dom(φ^{−1}) , \\ 0 & otherwise, \end{cases}

for every ξ ∈ L^{2}(R, L^{2}(P)). One checks that W_{φ}W_{ψ} = W_{φ∘ψ} and W_{φ}^{∗} = W_{φ^{−1}}. We represent L^{∞}(X) \overline{⊗} P = L^{∞}(X, P) on L^{2}(R, L^{2}(P)) by

(F ξ)(x, y) = F (x)ξ(x, y) for all ξ ∈ L^{2}(R, L^{2}(P)) and F ∈ L^{∞}(X, P) .

Note that the partial isometries W_{φ}, φ ∈ [[R]], normalize L^{∞}(X, P) and that

(W_{φ}^{∗}F W_{φ})(x) = \begin{cases}α_{ω(x,φ(x))}(F (φ(x))) & if x ∈ dom φ \\ 0 & otherwise. \end{cases}

Define M as the von Neumann algebra generated by L^{∞}(X, P) and the partial isometries W_{φ}, φ ∈ [[R]]. Denoting by Δ ⊂ R the diagonal subset, the orthogonal projection onto L^{2}(Δ, L^{2}(P)) implements a normal faithful conditional expectation E : M → L^{∞}(X) \overline{⊗} P satisfying

E(W_{φ}) = χ_{{x|φ(x)=x}} ⊗ 1 for all φ ∈ [[R]] .

The formula Tr := (μ ⊗ Tr) ∘ E defines a normal semifinite faithful trace on M. Fix a nonzero projection q ∈ P with Tr(q) = 1. Define the projection p ∈ L^{∞}(X) \overline{⊗} P given by p = 1 ⊗ q. Write A := L^{∞}(X)p and M := pMp. Then A is a quasi-regular abelian von Neumann subalgebra of M and the restriction of Tr to M gives a normal faithful tracial state τ on M. The relative commutant L^{∞}(X)′ ∩ M equals L^{∞}(X) \overline{⊗} P. Since P is a factor, it follows that Z(A′ ∩ M) = A. We finally prove that R ≅ R(A ⊂ M). Write R = ⋃_{k} graph(φ_{k}), with φ_{k} ∈ [R]. Then φ_{k} induces an automorphism of L^{∞}(X) and hence of A = L^{∞}(X)p that we denote by β_{k} ∈ Aut(A). Since P is a II_{∞} factor and q ∈ P is a finite projection, we can choose partial isometries w_{n} ∈ P such that ∑_{n} w_{n}^{∗}w_{n} = 1 and w_{n}w_{n}^{∗} = q for all n. Define the elements

v_{n,k} := (1 ⊗ w_{n})W_{φ_{k}}p .

All v_{n,k} belong to Q_{M}(A) and α_{v_{n,k}} equals the restriction of β_{k} to Ap_{n,k} for projections p_{n,k} ∈ A. Since the sum of all w_{n}^{∗}w_{n} equals 1, we also have that ⋁_{n} p_{n,k} = p. Therefore the graphs of the partial automorphisms α_{v_{n,k}} generate an equivalence relation that is isomorphic with R. To conclude the proof, we put F := {v_{n,k} | n, k ∈ ℕ} and observe that M = (F ∪ F^{∗} ∪ (A′ ∩ M))′′. By Lemma 3.4, the equivalence relation R(A ⊂ M) is generated by the graphs of the partial automorphisms α_{v_{n,k}}.

4. Proof of Theorem A

Throughout this section, we assume that n and m are integers satisfying 2 ≤ n < |m|. As explained in the introduction, the corresponding groups BS(n, m) form a complete list of the nonamenable icc Baumslag-Solitar groups up to isomorphism. Throughout this section, we write M = L(BS(n, m)) and A = {u_{a}, u_{a}^{∗}}′′. We start with the following observation.

Proposition 4.1. We have that A ⊂ M is a quasi-regular abelian von Neumann subalgebra satisfying Z(A′∩M) = A. Moreover, A′∩M has no amenable direct summand.

Proof. It is clear that A ⊂ M is a quasi-regular abelian von Neumann subalgebra, because the element a ∈ BS(n, m) generates an almost normal abelian subgroup of BS(n, m) : for every g ∈ BS(n, m), the group ga^{ℤ}g^{−1} ∩ a^{ℤ} has finite index in a^{ℤ}. To prove that Z(A′ ∩ M) = A, we define the finite index subalgebra A_{0} := {u_{a}^{n}, u_{a}^{−n}}′′ of A. We will first prove that Z(A′_{0} ∩ M) = A_{0}. Afterwards we will show that this implies that Z(A′ ∩ M) = A. Define G := ⟨a^{ℤ}, b^{−1}a^{ℤ}b⟩ ⊂ BS(n, m). Then L(G) is a subalgebra of A′_{0} ∩ M. So Z(A′_{0} ∩ M) ⊂ L(G)′ ∩ M. Using Lemma 2.1, one can easily see that {gγg^{−1} | g ∈ G} is an infinite set for every γ ∈ BS(n, m) − a^{nℤ}. Therefore L(G)′ ∩ M ⊂ A_{0}. This shows that Z(A′_{0} ∩ M) ⊂ A_{0}. Since the converse inclusion is obvious, we find that A_{0} = Z(A′_{0} ∩ M).

Since A_{0} ⊂ A has finite index, there exist orthogonal projections p_{j} ∈ A such that Ap_{j} = A_{0}p_{j} and ∑_{j} p_{j} = 1. But then

Z(A′ ∩ M)p_{j} = Z((A′ ∩ M)p_{j}) = Z((Ap_{j})′ ∩ p_{j}Mp_{j})
= Z((A_{0}p_{j})′ ∩ p_{j}Mp_{j}) = Z(p_{j}(A′_{0} ∩ M)p_{j})
= Z(A′_{0} ∩ M)p_{j} = A_{0}p_{j} ⊂ A

Therefore Z(A′ ∩ M) ⊂ A. The converse inclusion being obvious, we have proven that A = Z(A′ ∩ M). Using Lemma 2.1, it follows that G is an amalgamated free product of two copies of ℤ over a copy of ℤ embedded as nℤ and mℤ respectively. In particular, G is nonamenable and L(G) has no amenable direct summand. Since L(G) ⊂ A′_{0} ∩ M, it follows that A′_{0} ∩ M has no amenable direct summand either. As above, we have that

(A′ ∩ M)p_{j} = p_{j}(A′_{0} ∩ M)p_{j}

for all j. Hence A′ ∩ M has no amenable direct summand.

We now identify the associated countable equivalence relation R(A ⊂ M).

Proposition 4.2. The equivalence relation R(A ⊂ M) is isomorphic with the unique hyperfinite ergodic countable equivalence relation of type III_{n/|m|}.

Proof. Let k be the greatest common divisor of n and |m|. Write n = n_{0}k and m = m_{0}k. By our assumptions on n and m, we have that 1 ≤ n_{0} < |m_{0}|. Define the countable Borel equivalence relation R_{n,m} on the circle 𝕋 given by

R_{n,m} := {(y, z) ∈ 𝕋 × 𝕋 | ∃a, b ∈ ℕ such that a + b > 0

and y^{(n_{0}^{a}m_{0}^{b}k)} = z^{(m_{0}^{a}n_{0}^{b}k)}} .

Equip 𝕋 with its Lebesgue measure λ and note that R_{n,m} is a nonsingular countable Borel equivalence relation on (𝕋, λ). Define R_{0} := {(y, z) ∈ 𝕋 × 𝕋 | y^{m} = z^{n}}. Note that R_{0} ⊂ R_{n,m} and that R_{n,m} is the smallest equivalence relation containing R_{0}. Define π : R_{0} → 𝕋 : π(y, z) = y^{m}. Note that π is n|m|-to-1. Define the probability measure μ on R_{0} given by

μ(U) = \frac{1}{n|m|} ∫_{𝕋} #(U ∩ π^{−1}(\{x\})) dλ(x) .

For all k, l ∈ ℤ, we define the function P_{k,l} : R_{0} → 𝕋 : P_{k,l}(y, z) = y^{k}z^{l}. A direct computation yields a unique unitary

T : L^{2}(R_{0}, μ) → \overline{span}^{||·||_{2}}Au_{b}A : P_{k,l} ↦ u_{a}^{k}u_{b}u_{a}^{l} .

We turn L^{2}(R_{0}, μ) into an L^{∞}(𝕋)-L^{∞}(𝕋)-bimodule by the formula

(F · ξ · F′)(y, z) = F (y) ξ(y, z) F′(z) .

Under the natural identification of L^{∞}(𝕋) and A, the unitary T is A-A-bimodular. By construction _{A}L^{2}(R_{0}, μ)_{A} is isomorphic with a direct sum of bimodules of the form _{A}H(α_{j})_{A} where the union of the graphs of the partial automorphisms α_{j} equals R_{0} and hence generates the equivalence relation R_{n,m}. Applying Lemma 3.4 to F = {u_{b}}, we conclude that R(A ⊂ M) ≅ R_{n,m} up to measure zero. As in Section 2.4, denote by ω : R_{n,m} → ℝ the Radon-Nikodym 1-cocycle. Denote by Λ ⊂ 𝕋 the subgroup given by

Λ := \{exp(\frac{2πis}{(n_{0}m_{0})^{b}} ) | s ∈ ℤ, b ∈ ℕ\} .

For every z ∈ Λ, we denote by α_{z} : 𝕋 → 𝕋 the rotation α_{z}(y) = zy. We have graph α_{z} ⊂ R_{n,m} for all z ∈ Λ. Since all α_{z} are measure preserving, we actually have graph α_{z} ⊂ Ker(ω). Since Λ ⊂ 𝕋 is a dense subgroup, it follows that Ker(ω) is an ergodic equivalence relation. In particular, R_{n,m} is ergodic. A direct computation shows that ω(y, z) = log(n/|m|) for all (y, z) ∈ R_{0}. Since R_{0} generates the equivalence relation R_{n,m}, it follows that the essential image of ω equals log(n/|m|)ℤ. Using Lemma 2.7, we conclude that R_{n,m} is of type III_{n/|m|}. By construction, R_{n,m} is amenable and hence, hyperfinite.

We are now ready to prove our main theorem.

Proof of Theorem A. Fix for i = 1, 2, integers n_{i}, m_{i} ∈ ℤ with 2 ≤ n_{i} < |m_{i}|. Put M_{i} = L(BS(n_{i}, m_{i})) and denote by A_{i} ⊂ M_{i} the abelian von Neumann subalgebra generated by u_{a}, where a ∈ BS(n_{i}, m_{i}) is the first canonical generator. Assume that M_{1} and M_{2} are stably isomorphic. We must prove that

\frac{n_{1}}{|m_{1}|} = \frac{n_{2}}{|m_{2}|} .  (4)

Interchanging if necessary the roles of M_{1} and M_{2}, we can take a nonzero projection p_{1} ∈ A_{1} and a ∗-isomorphism α : p_{1}M_{1}p_{1} → M_{2}. We claim that inside M_{2}, we have α(A_{1}p_{1}) ≺ A_{2}. From Proposition 4.1, we know that

P := α(A_{1}p_{1})′ ∩ M_{2} = α((A′_{1} ∩ M_{1})p_{1})

has no amenable direct summand. By Proposition 3.1 in [Ue07], the HNN extension M_{2} can be viewed as the corner of an amalgamated free product of tracial von Neumann algebras. Since P has no amenable direct summand, it then follows from [CH08, Theorem 4.2] that P′ ∩ M_{2} ≺ A_{2}. So our claim that α(A_{1}p_{1}) ≺ A_{2} follows. By symmetry, we also have the intertwining α^{−1}(A_{2}) ≺ A_{1}p_{1} inside p_{1}M_{1}p_{1}. Applying α, we find that A_{2} ≺ α(A_{1}p_{1}) inside M_{2}. Having proven that inside M_{2} we have the intertwining relations α(A_{1}p_{1}) ≺ A_{2} and A_{2} ≺ α(A_{1}p_{1}), it follows from Theorem 3.3 that the equivalence relations R(A_{1}p_{1} ⊂ p_{1}M_{1}p_{1}) and R(A_{2} ⊂ M_{2}) are stably isomorphic. By construction, R(A_{1}p_{1} ⊂ p_{1}M_{1}p_{1}) is the restriction of R(A_{1} ⊂ M_{1}) to the support of p_{1}. So we conclude that the equivalence relations R(A_{1} ⊂ M_{1}) and R(A_{2} ⊂ M_{2}) are stably isomorphic. In particular, these ergodic nonsingular equivalence relations must have the same type. Using Proposition 4.2, we find (4).

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Niels Meesschaert KU Leuven Department of Mathematics Leuven (Belgium) niels.meesschaert@wis.kuleuven.be Stefaan Vaes KU Leuven Department of Mathematics Leuven (Belgium) stefaan.vaes@wis.kuleuven.be 646

Documenta Mathematica 19 (2014)