ON A CLASS OF TYPE II_{1} FACTORS

WITH BETTI NUMBERS INVARIANTS

SORIN POPA

University of California, Los Angeles

Abstract. We prove that a type II_{1} factor M has at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. In particular, this shows that the Betti numbers of A ⊂ M, as defined in ([G2]), are isomorphism invariants for the factors M having such Cartan subalgebras. Examples of factors with this property are the group measure space algebras associated with certain actions of the free groups with finitely many generators on the probability space, for which the Betti numbers are calculated in ([G2]). As a consequence, we obtain the first examples of type II_{1} factors M with trivial fundamental group and solve a problem formulated in ([K1]). Also, our results bring some insight into a recent problem of A. Connes.

0. Introduction.

We consider in this paper the class of type II_{1} factors M having maximal abelian von Neumann subalgebras A with normalizer acting ergodically on it and so that A ⊂ M satisfies both a rigidity property (in the spirit of Kazhdan and Connes-Jones, [Kaz], [CJ1]) and a compact approximation property (in the spirit of Haagerup, [H]). We call HT the class of all these factors.

We prove that if M ∈ HT , then any two such maximal abelian subalgebras of M are conjugate by a unitary element of M. More generally, any isomorphism between factors M ∈ HT must carry the corresponding maximal abelian ∗-subalgebras A one onto the other (modulo perturbation by an inner automorphism). As a result, for factors M ∈ HT , any isomorphism invariant for the inclusion A ⊂ M becomes an isomorphism invariant for M.

Typeset by AMS-TEX

Supported in part by a NSF Grant 9801324.

In particular, invariants for the measure-preserving countable orbit equivalence relation R_{M} induced on the standard probability space (X, μ) (for some identification A ≃ L^{∞}(X, μ)) by the normalizer of A in M, are invariants for M ∈ HT .

In a recent paper, D. Gaboriau introduced a notion of ℓ^{2} Betti numbers for arbitrary countable orbit equivalence relations R, {β_{n}(R)}_{n≥0}, starting from ideas of Atiyah ([A]) and Connes ([C4]), which generalizes the notion of Betti numbers for measurable foliations in ([C4]). This notion also generalizes the more combinatorial ℓ^{2} Betti numbers for discrete groups of Cheeger-Gromov ([ChGr]), β_{n}(G), for which he shows that β_{n}(G) = β_{n}(R_{G}), for any countable equivalence relation R_{G} implemented by a free, ergodic, measure-preserving action of the group G on a standard probability space (X, μ) ([G2]).

We define the Betti numbers {β_{n}(M)}_{n≥0} of a factor M in the class HT as the Betti ℓ^{2}- numbers of the corresponding orbit equivalence relation R_{M}, {β_{n}(R_{M})}_{n}.

Some of the general properties of the Betti numbers for countable equivalence relations proved in ([G2]) easily entail similar properties for the Betti numbers of the factors in the class HT . For instance, noticing that HT is closed under amplifications by arbitrary t > 0, it follows from ([G2]) that β_{n}(M^{t}) = β_{n}(M)/t, ∀n. Also, HT is closed under tensor products and a Künneth type formula holds for β_{n}(M_{1} ⊗ M_{2}) in terms of the Betti numbers for M_{1}, M_{2} ∈ HT , as a result of the similar formula for equivalence relations in Gaboriau’s paper ([G2]).

Our main example of a factor in the class HT is the group von Neumann algebra L(G) associated with G = ℤ^{2} ⋊ SL(2, ℤ), regarded as the group-measure space construction L^{∞}(𝕋^{2}, μ) = A_{0} ⊂ A_{0} ⋊_{σ_{0}} SL(2, ℤ). More generally, we consider cross product factors of the form A ⋊_{σ} SL(2, ℤ) with A abelian and σ a product of the action σ_{0} with other ergodic actions of SL(2, ℤ). By a recent result of G. Hjorth ([Hj]), based on the notion and results on tree-ability in ([G1]), these factors are in fact amplifications of group-measure space factors of the form L^{∞}(X, μ) ⋊ 𝔽_{n}, where 𝔽_{n} is the free group on n generators, n = 2, 3, ....

To prove that M belongs to the class HT , with A its “distinguished” maximal abelian subalgebra, we use Kazhdan’s rigidity of the inclusion ℤ^{2} ⊂ ℤ^{2} ⋊ SL(2, ℤ) and Haagerup’s compact approximation property of SL(2, ℤ). Note that in this case the algebra A is in fact a Cartan subalgebra of M, i.e., the normalizer of A in M generates all the von Neumann algebra M.

Since the orbit equivalence relation R_{M} implemented by SL(2, ℤ) on A has nontrivial Betti numbers (cf. [G1, 2]), it follows that the factors M = A ⋊_{σ} SL(2, ℤ) have non-trivial Betti numbers.

Other examples of factors in the class HT are obtained by taking discrete groups Γ that can be embedded as lattices in SU(n, 1) or SO(n, 1) and suitable actions σ of such Γ on abelian von Neumann algebras A ≃ L(ℤ^{N}). Indeed, these groups Γ have the Haagerup approximation property by ([dCaH], [CowH]) and their action on A can be taken to be rigid by ([Val]).

As a consequence of these considerations, we are able to answer a number of open questions in the theory of type II_{1} factors. Thus, the factors M = A ⋊_{σ} SL(2, ℤ) (more generally, A⋊_{σ} Γ with Γ, σ as above) provide the first class of type II_{1} factors with trivial fundamental group, i.e. F(M) =^{def} {t > 0 | M^{t} ≃ M} = {1}. Indeed, we mentioned that β_{n}(M^{t}) = β_{n}(M)/t, ∀n, so that if β_{n}(M) ≠ 0 for some n then F(M) = {1}.

In particular, the factors M are not isomorphic to the algebra of n by n matrices over M, for any n ≥ 2 (see Kadison’s Problem 3 in [K1]). Also, through appropriate choice of the action σ, we obtain factors of the form M ⋊_{σ} SL(2, ℤ) having the property Γ of Murray and von Neumann, yet trivial fundamental group.

The fundamental group F(M) of a II_{1} factor M was defined by Murray and von Neumann in the early 40’s, in connection with their notion of continuous dimension. They noticed that F(M) = ℝ_{+}^{∗} when M is isomorphic to the hyperfinite type II_{1} factor R, and more generally when M “splits off” R.

The first examples of type II_{1} factors M with F(M) ≠ ℝ_{+}^{∗}, and the first occurence of rigidity in the von Neumann algebra context, were discovered by Connes in ([C1]). He proved that if G is an infinite conjugacy class discrete group with the property T of Kazhdan then its group von Neumann algebra M = L(G) is a type II_{1} factor with countable fundamental group. It was then proved in ([Po1]) that this is still the case for factors M which contain some irreducible copy of L(G). It was also shown that there exist type II_{1} factors M with F(M) countable and containing any presribed countable set of numbers ([Gol], [Po3, 5]). However, the fundamental group F(M) could never be computed exactly, in any of these examples.

In fact, more than proving that F(M) = {1} for M = A ⋊_{σ} SL(2, ℤ), the calculation of the Betti numbers shows that M^{t_{1}} ⊗ M^{t_{2}}... ⊗ M^{t_{n}} is isomorphic to M^{s_{1}} ⊗M^{s_{2}}...⊗M^{s_{m}} if and only if n = m and t_{1}t_{2}...t_{n} = s_{1}s_{2}...s_{m}. In particular, all tensor powers of M, M^{⊗^{n}}, n = 2, 3, ..., are mutually non-isomorphic. (N.B. The first examples of factors satisfying this latter property were constructed in [C4]; another class of examples was obtained in [CowH]). Moreover, since M^{t} ≃ L^{∞}(X, μ)⋊𝔽_{n} for t = (12(n−1))^{−1} (cf. [Hj]), it follows that there exists a free ergodic action σ_{n} of 𝔽_{n} on the standard probability space (X, μ) such that M_{n} = L^{∞}(X, μ) ⋊_{σ_{n}} 𝔽_{n} satisfy M_{k_{1}} ⊗ ... ⊗ M_{k_{n}} ≃ M_{l_{1}} ⊗ ... ⊗ M_{l_{m}} if and only if n = m and k_{1}k_{2}...k_{n} = l_{1}l_{2}...l_{m}. Moreover, in case n = m, the factors are stably isomorphic for any choice of k_{i}, l_{j}. Similar results hold true in the case of lattices Γ in SO(n, 1) or in SU(n, 1). These classes of factors are non- stably equivalent, for different values of n.

We also prove that the factors M = A⋊_{σ} SL(2, ℤ), as well as any tensor product of amplifications of M, have only subfactors of integer Jones index, if finite. In particular, they have no subfactors with values in the Jones series 4cos^{2}π/n, except for n = 3, 4, 6.

As for the “size” of HT , we mention that we do not know of other examples than the ones mentioned above, namely the factors M of the form A ⋊ G, with G = SL(2, ℤ) or G a lattice in SU(n, 1), SO(n, 1), their amplifications, tensor products and certain cross-products. The class HT does not contain the hyperfinite type II_{1} factor. The factors M ∈ HT cannot contain property T factors and cannot be embedded into free group factors (due to arguments similar to [CJ1]).

Besides these concrete applications, our results give a partial answer to a challenging problem recently raised by Alain Connes, on defining a notion of Betti numbers for type II_{1} factors, from similar conceptual grounds as in the case of orbit equivalence relations (simplicial structure, L^{2}-cohomology, etc). In this respect, our definition is only for the special class of factors HT and falls short of being conceptual in the above sense. Yet it already produces significant applications and emphasizes phenomena that could be used as a test case for the general problem.

The paper is organized as follows: In Section 1 we prove a key perturbation result for maximal abelian von Neumann subalgebras (hereafter abbreviated as m.a.s.a.) in type II_{1} factors. The proof uses Christensen’s set-up and techniques ([Chr]), and a careful analysis of the geometry of projections. In Section 2 we consider a relative version of Haagerup’s compact approximation property. In Section 3 we use the Connes-Jones idea of translating rigidity in terms of properties of completely positive maps ([CJ1]), to define the notion of rigid inclusion (or property T pair) of algebras, by analogy with the case of groups.

In Section 4 we define the class HT of factors M having m.a.s.a.’s A with the property that A ⊂ M is rigid and M has the Haagerup property relative to A. We prove the main technical result of the paper, showing that such m.a.s.a.’s are unique. We also prove the stability of HT with respect to various operations (amplification, tensor product, finite index induction/restriction, etc). In Section 5 we construct examples of such factors. In Section 6 we define the Betti numbers {β_{n}(M)}_{n} for M ∈ HT and use ([G2]) to deduce various properties of this invariant. We end with applications, as well as some remarks and open questions.

Acknowledgement. It is a pleasure for me to thank Alain Connes, Uffe Haagerup, Dima Shlyakhtenko and Alain Valette for many fruitful discussions and invaluable comments, as well as Damien Gaboriau for keeping me informed on his beautiful recent results. I am very greatful to MSRI and the organizers of the Operator Algebra year 2000-2001, for their hospitality and for a most stimulating atmosphere.

1. A perturbation result.

In this Section we prove a perturbation result concerning abelian von Neumann subalgebras of type II_{1} factors. This result will play a key role in the proof of Theorem 4.1, which is the main technical result in this paper.

The proof uses the perturbation techniques from [Chr], the “pull down” identity ([PiPo]) and a careful handling of the geometry of projections, including a result from ([K2]).

Since we will also use it in the next Section, let us recall from ([Chr], [J1], [Po2]) some well known facts about the basic construction for an inclusion of a von Neumann algebra B into a type II_{1} factor N. Thus, we denote by ⟨N, B⟩ the von Neumann algebra generated in B(L^{2}(N, τ)) by N (regarded as the algebra of left multiplication operators by elements in N) and by the orthogonal projection e_{B} of L^{2}(M, τ) onto L^{2}(B, τ).

Since e_{B}xe_{B} = E_{B}(x)e_{B}, ∀x ∈ N, where E_{B} is the unique τ-preserving conditional expectation of N onto B, it follows that spNe_{B}N is a *-algebra with support equal to 1 in B(L^{2}(N, τ)). Thus, ⟨N, B⟩ = \overline{sp}^{w}\{xe_{B}y | x, y ∈ N\} and e_{B}⟨N, B, ⟩e_{B} = Be_{B}.

One can also readily see that if J denotes the canonical conjugation on the Hilbert space L^{2}(N, τ), then ⟨N, B⟩ = JBJ′. This shows in particular that ⟨N, B⟩ is semifinite.

We endow ⟨N, B⟩ with the unique normal semifinite faithful trace T r satisfying T r(xe_{B}y) = τ(xy), ∀x, y ∈ N. Note that there exists a unique N bilinear map Φ from spNe_{B}N ⊂ ⟨N, B⟩ into N satisfying Φ(xey) = xy, ∀x, y ∈ N, and τ ∘ Φ = T r. The map Φ extends uniquely to a N-bilinear map from L^{1}(⟨N, B⟩, T r) onto L^{1}(N, τ), still denoted Φ. This bilinear map satisfies the “pull down” identity eX = eΦ(eX), ∀X ∈ ⟨N, B⟩ (see [PiPo], or [Po7]). Note that Φ(eX) actually belongs to L^{2}(N, τ) ⊂ L^{1}(N, τ), for X ∈ ⟨N, B⟩.

Let now U_{0} ⊂ N be a subgroup of unitary elements and B_{0} = U′′_{0} the von Neumann algebra it generates. If b is an element in ⟨N, B⟩ with T r(b^{∗}b) < ∞ then we denote by K_{U_{0}}(b) = \overline{co}^{w}\{u_{0}bu_{0}^{∗} | u_{0} ∈ U_{0}\}. Note that K_{U_{0}}(b) is also contained in the Hilbert space L^{2}(⟨N, B⟩, T r), where it is still weakly closed.

Let h = h_{U_{0}}(b) ∈ K_{U_{0}}(b) be the unique element of minimal norm ∥ ∥_{2,T r} in K_{U_{0}}(b). Since uK_{U_{0}}(b)u^{∗} = K_{U_{0}}(b) and ∥uhu^{∗}∥_{2,T r} = ∥h∥_{2,T r}, ∀u ∈ U_{0}, by the uniqueness of h it follows that uhu^{∗} = h, ∀u ∈ U_{0}. Thus h ∈ U′_{0} ∩ ⟨N, B⟩ = B′_{0} ∩ ⟨N, B⟩. Moreover, by the definitions, we see that if 0 ≤ b ≤ 1 then 0 ≤ k ≤ 1 for all k ∈ K_{U_{0}}(b).

1.1. Theorem. Assume B is a maximal abelian ∗-subalgebra in N and B_{0} is an abelian von Neumann subalgebra such that B_{01} =^{def} B′_{0} ∩ N is abelian (thus maximal abelian). Then the following conditions are equivalent:

1^{∘}. The element h_{U_{0}}(e_{B}), of minimal norm ∥ ∥_{2,T r} in K_{U_{0}}(e_{B}), is non-zero and belongs to B′_{0} ∩ ⟨N, B⟩.

2^{∘}. There exists a non-zero projection e_{0} ∈ B′_{0} ∩ ⟨N, B⟩ with T r(e_{0}) < ∞.

3^{∘}. There exist non-zero projections q_{0} ∈ B′_{0} ∩ N, q ∈ B and a partial isometry v ∈ N such that v^{∗}v = q_{0}, vv^{∗} = q and vB_{0}v^{∗} ⊂ Bq.

Proof. 3^{∘} ⟹ 1^{∘}. If v satisfies condition 3^{∘} then B_{0}q_{0} is contained in v^{∗}Bv. Thus, e_{0} = v^{∗}e_{B}v belongs to B′_{0} ∩ ⟨N, B⟩ and has finite trace.

1^{∘} ⟹ 2^{∘}. Let e_{0} be the spectral projection of h = h_{U_{0}}(e_{B}) corresponding to the interval (∥h∥/2, ∞). Then e_{0} ≠ 0 and h ≥ \frac{1}{2}e_{0}. Thus,

T r(e_{0}) ≤ 2T r(h) ≤ 2T r(e_{B}) < ∞.

Thus, e_{0} is a finite projection in N_{1} = ⟨N, B⟩ and e_{0} commutes with B_{0} (since h does).

2^{∘} ⟹ 3^{∘}. Since B_{0}e_{0} is abelian, it is contained in a maximal abelian subalgebra B_{1} of e_{0}N_{1}e_{0}.

By a result of Kadison ([K2]), B_{1} contains a non-zero abelian projection e_{1} of N_{1} (i.e., e_{1}N_{1}e_{1} is abelian). Since e_{B} has central valued (semifinite) trace equal to 1 (it is also a maximal abelian projection in N_{1}), it follows that e majorizes e_{1}.

Let V ∈ N_{1} be a partial isometry such that V^{∗}V = e_{1} ≤ e_{0} and V V^{∗} ≤ e_{B}. Moreover V B_{1}e_{1}V^{∗} is a subalgebra of e_{B}N_{1}e_{B} = Be. Since e_{1} commutes with B_{0}, it follows that if we denote by f′ the maximal projection in B_{0} such that f′e_{1} = 0 and let f_{0} = 1 − f′, then there exists a unique isomorphism φ from B_{0}f_{0} into B such that φ(b)e_{B} = V bV^{∗}, ∀b ∈ B_{0}f. Let f = φ(f_{0}) ∈ B.

It follows that φ(b)e_{B}V = e_{B}V b, ∀b ∈ B_{0}f_{0}. By applying Φ to both sides and denoting a the square integrable operator a = Φ(eV) ∈ L^{2}(N, τ), it follows that φ(b)a = ab, ∀b ∈ B_{0}. Since ea = eV = V, it follows that a ≠ 0.

By the usual trick, if we denote by v_{0} ∈ N the unique partial isometry in the polar decomposition of a such that the right supports of a and v_{0} coincide, then p_{0} = v_{0}^{∗}v_{0} belongs to the algebra B′_{0} ∩ N = B_{01}, which is abelian by hypothesis, p = v_{0}v_{0}^{∗} belongs to (φ(B_{0})f)′ ∩ fNf and φ(b)v_{0} = v_{0}b, ∀b ∈ B_{0}f_{0}.

But B_{01} = B′_{0} ∩ N maximal abelian in N implies B_{01}f_{0} maximal abelian in f_{0}Nf_{0}. Noreover, since v_{0}B_{0}v_{0}^{∗} = φ(B_{0})p, if we denote B_{11} = v_{0}B_{01}v_{0}^{∗}, then by spatiality we have

B_{11} = v_{0}B_{01}v_{0}^{∗} = v_{0}(B′_{0} ∩ N)v_{0}^{∗}
= v_{0}B_{0}v_{0}^{∗}′ ∩ pNp = φ(B_{0})p′ ∩ pNp.

Thus, the commutant of φ(B_{0})p is abelian. Since any element in pBp commutes with φ(B_{0})p, it follows that pBp ⊂ (φ(B_{0})p)′ ∩ pNp = B_{11} ⊂ (φ(B_{0})f)′ ∩ fNf. Moreover, (p(φ(B_{0})f)′ ∩ fNf)p = pBp ⊂ (φ(B_{0})p)′ ∩ pNp = B_{11}.

It follows that if P_{1} denotes the von Neumann algebra generated by p and (φ(B_{0})f)′ ∩ fNf ⊃ Bf inside fNf, then pP_{1}p = (p(φ(B_{0})f)′ ∩ fNf)p = B_{11} is abelian.

Thus, p is an abelian projection in P_{1}. Since P_{1} is a finite von Neumann algebra, there exists a central projection z_{1} of P_{1}, under the central support of p in P_{1}, such that pz_{1} ≠ 0 and such that P_{1}z_{1} is homogeneous of type n, for some n ≥ 1. Note that the center of P_{1} is included in Bf the latter being maximal abelian in fNf, thus in P_{1} ⊂ fNf. Thus, z_{1} ∈ Bf.

Now, since pz_{1} has central trace equal to 1/n in P_{1}z_{1} and Bz_{1} is maximal abelian in P_{1}z_{1}, it follows that there exists a projection f_{11} ∈ Bf such that f_{11} is equivalent to p_{1}z_{1} in P_{1}z_{1} (see [K2]). Let v_{1} ∈ P_{1}z_{1} be such that v_{1}v_{1}^{∗} = f_{11}, v_{1}^{∗}v_{1} = p_{1}z_{1}. Since p_{1}z_{1} is abelian in P_{1}, f_{11} is also abelian, thus f_{11}P_{1}f_{11} = Bf_{11}. This implies that v_{1}pz_{1}(φ(B_{0})′ ∩ fNf))pz_{1}v_{1}^{∗} = Bf_{11}.

Finally, since pz_{1} ≤ p, φ(B_{0})p = v_{0}B_{0}v_{0}^{∗} and pz_{1} belongs to φ(B_{0})′, it follows that v_{0}^{∗}pz_{1}v_{0} ∈ B′_{0}. This shows that if we denote v = v_{1}v_{0}, then v is a partial isometry satsifying v^{∗}v ∈ B′_{0}, vv^{∗} = f_{11} ∈ B and vB_{0}v^{∗} ⊂ Bvv^{∗}. Q.E.D.

In Section 4 we will in fact need a consequence of Theorem 1.1. To state it, recall from ([D1]) that a maximal abelian ^{∗}-subalgebra (abreviated m.a.s.a.) B of a von Neumann factor N is called semiregular if the set of unitaries of N that normalize B, i.e., N (B) = {u ∈ U(N) | uBu^{∗} = B}, generate a factor, equivalently, N (B)′∩N = ℂ. Also, B is called regular if this normalizer generetes all the ambient factor N. Such regular MASA’s were later called Cartan subalgebras in ([FM]), a terminology that seems to have prevailed and which we will therefore adopt.

1.2. Corollary. Let N be an arbitrary type II_{1} factor. Let B be a semiregular m.a.s.a. of N and B_{0} ⊂ N an abelian von Neumann subalgebra such that B′_{0}∩N is a semiregular m.a.s.a. of N. If B, B_{0} satisfy the equivalent conditions in Theorem 1.1 then there exists a unitary element u ∈ N such that u(B′_{0}∩N)u^{∗} = B. In particular, this is the case if B, B′_{0} ∩ N are Cartan subalgebras of N.

Proof. By Theorem 1.1 there exists a non-zero partial isometry v ∈ N such that v^{∗}v ∈ B′_{0} ∩ N, vv^{∗} ∈ B, vB_{0}v^{∗} = Bvv^{∗}. Thus, v(B′_{0} ∩ N)v^{∗} = vB_{0}v^{∗}′ ∩ vv^{∗}Nvv^{∗} contains Bvv^{∗}′ ∩ vv^{∗}Nvv^{∗} = Bvv^{∗}. Since both are maximal abelian, we get v(B′_{0} ∩ N)v^{∗} = Bvv^{∗}.

Moreover, by cutting v from the right with a smaller projection in B′_{0} ∩ N, we may clearly assume τ(vv^{∗}) = 1/n for some integer n.

Since B, B′_{0} ∩ N are semiregular, there exist partial isometries v_{1}, v_{2}, ..., v_{n}, respectively w_{1}, w_{2}, ..., w_{n} in the normalizing groupoids of B respectively B′_{0} ∩ N such that Σ_{i}v_{i}v_{i}^{∗} = 1, Σ_{j}w_{j}w_{j}^{∗} = 1 and v_{i}^{∗}v_{i} = v^{∗}v, w_{j}^{∗}w_{j} = vv^{∗}, ∀i, j. But then u = Σ_{i}w_{i}vv_{i}^{∗} is a unitary element and u(B′_{0} ∩ N)u^{∗} = B. Q.E.D.

2. Relative Haagerup property.

Recall that a type II_{1} factor N has Haagerup’s compact approximation property (property H hereafter) if there exists a sequence of compact completely positive, unital, trace preserving maps Φ_{n} : N → N such that

lim_{n→∞}∥Φ_{n}(x) − x∥_{2} = 0, ∀x ∈ N.

In this paper we need to introduce a relative version of this property. To this end, let us first note the following:

2.1. Lemma. Let N be a type II_{1} factor and B ⊂ N a von Neumann subalgebra. Let Φ : N → N be a unital, trace preserving, completely positive map satisfying

Φ(b_{1}xb_{2}) = b_{1}Φ(x)b_{2}, ∀x ∈ N, b_{1}, b_{2} ∈ B.

Then there exists a unique bounded operator T_{Φ} on the Hilbert space L^{2}(N, τ) such that T_{Φ}(x) = Φ(x), for x ∈ N. Moreover this operator lies in B′ ∩ ⟨N, B⟩.

Proof. By Kadison’s inequality, for x ∈ M we have

⟨T_{Φ}(x̂), T_{Φ}(x̂)⟩ = τ(Φ(x)^{∗}Φ(x)) ≤ τ(Φ(x^{∗}x)) = τ(x^{∗}x), ∀x ∈ N.

Thus, ∥T_{Φ}∥ ≤ 1.

If b ∈ B is regarded as operator of left multiplication by b on L^{2}(N, τ), then we have

bT_{Φ}(x̂) = bΦ(x) = Φ(bx) = T_{Φ}(bx̂).

Thus, T_{Φ} ∈ B′.

Similarily,

JbJ(T_{Φ}(x̂)) = Φ(x)b = Φ(xb) = T_{Φ}(JbJ(x̂))

showing that T_{Φ} ∈ JBJ′ = ⟨N, B⟩. Q.E.D.

Note that if we take Φ to be the τ-preserving conditional expectation E_{B}, of N onto B, then T_{Φ} is the orthogonal projection e_{B} of L^{2}(N, τ). The next Lemma, which is an immediate extension of a well known result of Haagerup, gives examples of B-bimodular completely positive maps in the case of cross product inclusions.

2.2. Lemma. Let G be a discrete group and σ an action of G on the finite von Neumann algebra (B, τ) by τ-preserving automorphisms. Let N = B ⋊_{σ} G. Let φ : G → ℂ be a positive definite function on G with φ(e) = 1. Define M_{φ} on N by M_{φ}(Σ_{g}b_{g}u_{g}) = Σ_{g}φ(g)b_{g}u_{g}. Then M_{φ} is a unital, trace preseving, completely positive map and it is B-bilinear.

Proof. The proof of the case B = ℂ of (1.1 in [H]) works exactly the same in this case. Q.E.D.

We can now state the cross product version of Haagerup’s result. We will use the following:

2.3. Definition. 1^{∘}. A discrete group G has the property H if there exist positive definite functions ϕ_{n} on G such that ϕ_{n}(e) = 1, lim_{g→∞}ϕ_{n}(g) = 0, ∀n, and

lim_{n→∞}ϕ_{n}(g) = 1, ∀g ∈ G.

It should be noted that Gromov calls such groups G a-T-menable.

2^{∘}. Let N be a type II_{1} factor and B ⊂ N a von Neumann subalgebra. We say that N has the property H relative to B, or that B ⊂ N has the property H, if there exists a sequence of unital, completely positive, trace preserving maps Φ_{n} on N, with Φ(b_{1}xb_{1}) = b_{1}Φ(x)b_{2}, ∀x ∈ N, b_{1}, b_{2} ∈ B, such that

lim_{n→∞}∥Φ_{n}(x) − x∥_{2} = 0, ∀x ∈ M

and such that all the corresponding operators T_{Φ_{n}} defined in 2.1 belong to the ideal of compact operators of the semifinite von Neumann algebra ⟨N, B⟩.

Note that, following the above remark, one could also use the terminology: The factor N is a-T-menable relative to B.

2.4. Proposition. Let B, G, σ, N be like in Lemma 2.2. Then N has the property H relative to B if and only if G has the property H.

Proof. Straighforward. Q.E.D.

Let us recall some well known results from ([H], [dCaH], [CowH]), that we need later:

2.5. Proposition. Let B, G, σ, N be like in Lemma 2.2. Assume G is one of the following:

1^{∘}. G = SL(2, ℤ) or G = 𝔽_{n}, for some 2 ≤ n ≤ ∞,

2^{∘}. G is a discrete subgroup of SO(n, 1), for some n ≥ 2.

3^{∘}. G is a discrete subgroup of SU(n, 1), for some n ≥ 2.

4^{∘}. G = G_{1} ∗_{H} G_{2} for some discrete groups G_{1}, G_{2} with the property H, with H ⊂ G_{1}, H ⊂ G_{2} a common subgroup.

Then B ⊂ N has the relative property H.

Proof. Part 1^{∘} is a result from ([H]), then 2^{∘} is in ([dCaH]) and 3^{∘} in ([CowH]). Part 4^{∘} is also well known. Q.E.D.

We will also need the following simple observations:

2.6. Proposition. 1^{∘}. If B ⊂ N has the property H and p ∈ B is a projection, then pBp ⊂ pNp has the property H.

2^{∘}. If B_{0} ⊂ N_{0} and B_{1} ⊂ N_{1} have the property H then B_{0} ⊗ B_{1} ⊂ N_{0} ⊗ N_{1} has the property H.

3^{∘}. If B ⊂ N has the property H and B ⊂ N_{0} ⊂ N, then B ⊂ N_{0} has the property H.

4^{∘}. If B ⊂ B_{0} ⊂ N and B ⊂ B_{0} has a finite orthonormal basis then B_{0} ⊂ N has H implies B ⊂ N has H.

Proof. Immediate by the definitions. Q.E.D.

3. Rigid embeddings.

In this section we consider a notion of rigid m.a.s.a., and more generally of a rigid embedding of a von Neumann algebra into a type II_{1} factor, inspired by Kazhdan’s example of the embedding ℤ^{2} ⊂ ℤ^{2} ⋊ SL(2, ℤ). Thus, our definition will be the operator algebraic version of the de la Harpe-Valette notion of property T for pairs of groups ([dlHVa]), in the same spirit as the Connes-Jones definition of property T for von Neumann algebras ([CJ1]). To this end, we use Connes’ correspondences as a set-up, like in ([CJ1]).

First recall the definition for inclusions of groups:

3.1 Definition. An embedding of discrete groups H ⊂ G is called rigid if there exist finitely many elements g_{1}, g_{2}, ..., g_{n} ∈ G and ε > 0, such that if π : G → U(H) is a unitary representation of the group G on the Hilbert space H with a unit vector ξ ∈ H such that ∥π(g_{i})ξ − ξ∥ < ε, ∀i, then there exists a non-zero vector ξ_{0} ∈ H such that π(h)ξ_{0} = ξ_{0}, ∀h ∈ H. Note that in ([dHVa]), the terminology used is: the pair (G, H) has the property T.

The following equivalent characterisation of the condition 3.1 is well known and thus we omit the proof:

3.2. Lemma. H ⊂ G satisfies condition 3.1 if and only if it satisfies the following: There exist K ≥ 1, g_{1}, g_{2}, ..., g_{n} ∈ G and ε_{0} > 0, such that if π : G → U(H) is a unitary representation with a unit vector ξ ∈ H satisfying ∥π(g_{i})ξ − ξ∥ < δ, ∀i, for some δ ≤ ε_{0}, then ∥π(h)ξ − ξ∥ ≤ Kδ, ∀h ∈ H.

3.3. Definition. Let N be a finite von Neumann algebra and B ⊂ N a von Neumann subalgebra. We say that the embedding B ⊂ N is rigid (or that the pair (N, B) has the property T) if there exists a finite subset x_{1}, x_{2}, ..., x_{n} ∈ N and ε > 0 such that if H is a N bimodule with ξ ∈ H a unit vector such that ∥x_{i}ξ−ξx_{i}∥ ≤ ε, ∀i, then there exist a non-zero vector ξ_{0} ∈ H such that bξ_{0} = ξ_{0}b, ∀b ∈ B.

Note that when B = N this notion coincides with the notion of property T for von Neumann algebras of Connes and Jones in ([CJ1]).

3.4. Remark. In ([Po1]) we defined a notion of relative property T for inclusions B ⊂ N (or property T of N relative to B) which is complementary to the one above. This notion was also considered independently by Anantharam-Delaroche in ([A-De]). In the case of groups, the notion corresponds to the situation in which H ⊂ G is so that G is the semi-direct product of the group H by a property T group G_{0}. It should be noted that both the notion 3.3 considered here and the notion considered in ([Po1]) are in some sense “relative property T” for an inclusion B ⊂ N, but while the notion in ([Po1]) means “N has the property T relative to B”, the notion considered in this paper is a “property T of B relative to its embedding into N”.

3.5. Lemma. H ⊂ G is rigid if and only if L(H) ⊂ L(G) is rigid.

Proof. The proof is identical to the proof of the case H = G in ([CJ1]) and of the more general such statements in ([Po1]). Q.E.D.

3.6. Lemma. The following conditions are equivalent:

1^{∘}. B_{0} ⊂ N is rigid.

2^{∘}. ∃x_{1}, x_{2}, ..., x_{n} ∈ N, ε > 0, K ≥ 1, such that if δ > 0 and (H, ξ) is a Hilbert space with a unit vector satisfying ∥x_{i}ξ − ξx_{i}∥ ≤ δ, ∀i then there exists a vector ξ_{0} ∈ H such that ∥ξ_{0} − ξ∥ ≤ Kδ and bξ_{0} = ξ_{0}b, ∀b ∈ B_{0}.

3^{∘}. ∃x_{1}, x_{2}, ..., x_{n} ∈ N, ε > 0, K ≥ 1, such that if δ > 0 and ΦN → N is a normal, completely positive map with τ(Φ(1)) = 1 and ∥Φ(x_{i}) − x_{i}∥_{2} ≤ δ, ∀i, then ∥Φ(b) − b∥_{2} ≤ Kδ, ∀b ∈ B_{0}, ∥b∥ ≤ 1.

Proof. 1^{∘} ⟹ 2^{∘}. The proof is identical to the proof of 4.1.5 in ([Po1]).

2^{∘} ⟹ 3^{∘}. By using the equivalence between the bimodules and completely positive maps (Section 1.2 in [Po1]), this becomes a tautology (see also 4.2.1 in [Po1]).

3^{∘} ⟹ 1^{∘}. This is immediate from (1.2 in [Po1]). Q.E.D. The next result lists the main properties of rigid embeddings:

3.7. Proposition. 1^{∘}. If B_{0} ⊂ N is a rigid embedding and N ⊂ M then B_{0} ⊂ M is a rigid embedding. Conversely, if [M : N] < ∞ and B_{0} ⊂ M is a rigid embedding then B_{0} ⊂ N follows a rigid embedding.

2^{∘}. If B_{0} ⊂ N is a rigid embedding and B ⊂ B_{0} then B ⊂ N is a rigid embedding. Conversely, if B has a finite orthonormal basis with respect to B_{0} and B ⊂ N is a rigid embedding, then B_{0} ⊂ N is a rigid embedding.

3^{∘}. Let B ⊂ N and p ∈ P(B) a projection. Assume that there exist finitely many partial isometries v_{1}, v_{2}, ..., v_{n} ∈ N such that Σ_{i}v_{i}v_{i}^{∗} = 1 and v_{i}^{∗}v_{i} ∈ pBp. Then B ⊂ N is a rigid embedding if and only if pBp ⊂ pNp is a rigid embedding.

4^{∘}. If (B_{i} ⊂ N_{i}) are rigid embeddings for i = 1, 2 then (B_{1} ⊗ B_{2} ⊂ N_{1} ⊗ N_{2}) is a rigid embedding.

Proof. 1^{∘}. The first part is trivial. The second part has the same proof as (4.1.8 in [Po1]).

2^{∘}. The first part is trivial. The second part has same proof as (4.1.8 in [Po1]). 3^{∘}. If pBp ⊂ pNp is a rigid embedding then clearly B ⊂ N forllows rigid, by the

definitions, and by taking p to be in the critical set for B ⊂ N, with ε sufficiently small. The converse has the same proof as (4.1.9 in [Po1]).

4^{∘}. This has the same proof as analogue statements in ([Po1]). Q.E.D. Let us end this Section with some examples of rigid inclusions, following from

results of Kazhdan ([Kaz]) and respectively Valette ([Va]).

3.8. Proposition. 1^{∘}. The inclusion ℤ^{2} ⊂ Z^{2} ⋊ SL(2, ℤ) is rigid. Thus, L(ℤ^{2}) ⊂ L(ℤ^{2} ⋊ SL(2, ℤ)) is a rigid embedding of algebras.

2^{∘}. For each n ≥ 2 there exist free ergodic measure preserving actions σ of 𝔽_{n} on A ≃ L^{∞}(X, μ) such that A ⊂ A ⋊_{σ} 𝔽_{n} is rigid.

3^{∘}. For each n ≥ 2 there exist lattices Γ in SO(n, 1) (respectively in SU(n, 1)) which admit free ergodic measure preserving actions σ on A ≃ L^{∞}(X, μ) such that A ⊂ A ⋊_{σ} Γ is rigid.

Proof. Part 1^{∘} is a well known result in ([Kaz]). Part 2^{∘} then follows trivially from 1^{∘} and by the result of Hjorth, showing that if p ∈ L(ℤ^{2}) is a projection of trace (12(n − 1))^{−1} then pL(ℤ^{2})p ⊂ pL(ℤ^{2} ⋊ SL(2, ℤ^{2}))p is isomorphic to an inclusion of the form A ⊂ A ⋊_{σ} 𝔽_{n}, for some appropriate free, ergodic, measure preserving action σ of 𝔽_{n} on A ≃ L^{∞}(X, μ). The rigidity of this latter inclusion follows by property 3.7.3^{∘}.

Part 3^{∘} is a recent result of Alain Valette ([Va]). Q.E.D.

4. HT factors.

In this Section we prove the main technical result of this paper, showing that if a type II_{1} factor has Cartan subalgebras which have at the same time the property H (as considered in Sec. 2) and the relative property T (considered in Sec. 3), then such Cartan subalgebras are unique modulo conjugacy by unitary elements. More generally, we have:

4.1. Theorem. Let N be a type II_{1} factor with two abelian von Neumann subalgebras B, B_{0}. Assume that B′_{0} ∩ N is abelian and that both B and B′_{0} ∩ N are semiregular m.a.s.a.’s. Assume further that N has property H relative to B and that B_{0} ⊂ N is a rigid inclusion. Then there exists a unitary element u in N such that uB_{0}u^{∗} ⊂ B, and thus u(B′_{0} ∩ N)u^{∗} = B.

Proof. By Corollary 1.2, if we assume by contradiction that B′_{0} ∩ N and B are not unitary conjugate, then by Theorem 1.1, 0 ∈ K_{U_{0}}(e_{B}) ⊂ N_{1} = ⟨M, B⟩. This implies that given any finite projection f ∈ N_{1} and any ε > 0, there exists a unitary element u ∈ U_{0} such that T r(fue_{B}u^{∗}) < ε. Indeed, because if for some ε_{0} > 0 we would have T r(fue_{B}u^{∗}) ≥ ε_{0} then by taking appropriate convex combinations and weak limits, we would get that 0 = T r(f0) ≥ ε_{0}, a contradiction.

Let Φ_{n} : N → N be completely positive, unital trace preserving, B-bimodular maps tending strongly to the identity.

By the rigidity of the embedding B_{0} ⊂ N, there exists n large enough such that Φ = Φ_{n} satisfies

(1)  ∥Φ(v) − v∥_{2} ≤ 1/4, ∀v ∈ U(B_{0})

On the other hand, since T_{Φ} is in the compact ideal space of N_{1}, it follows that there exists a finite projection f ∈ N_{1} such that

(2)  ∥T_{Φ}(1 − f)∥_{2} < 1/4

Let then u ∈ U_{0} satisfy the condition

(3)  T r(fue_{B}u^{∗}) < 1/4

Let {m_{j}}_{j} ⊂ L^{2}(N, τ) be such that m_{j}e_{B}m_{j}^{∗} = f. Equivalently, ⊕_{j}L^{2}(m_{j}B) = fL^{2}(N, τ). Thus, if x ∈ N ⊂ L^{2}(N, τ) then f(x) = Σ_{j}m_{j}E_{B}(m_{j}^{∗}x) and ∥f(x)∥_{2}^{2} = Σ_{j}∥m_{j}E_{B}(m_{j}^{∗}x)∥_{2}^{2}.

It follows that we have:

T r(fue_{B}u^{∗}) = T r((Σ_{j}m_{j}e_{B}m_{j}^{∗})ue_{B}u^{∗}) = Σ_{j}τ(m_{j}E_{B}(m_{j}^{∗}u)u^{∗})
= Σ_{j}∥m_{j}E_{B}(m_{j}^{∗}x)∥_{2}^{2} = ∥f(u)∥_{2}^{2}

Taking this into account, by (2) and (3) we have:

∥T_{Φ}(u)∥_{2} ≤ ∥T_{Φ}((1 − f)(u))∥_{2} + ∥f(u)∥_{2}

(4)  < 3/4

But by (1), this implies:

∥u∥_{2} ≤ ∥T_{Φ}(u)∥_{2} + ∥Φ(u) − u∥_{2}

< 3/4 + 1/4 = 1.

Thus 1 = τ(uu^{∗}) < 1, a contradiction.

We conclude that the semiregular m.a.s.a’s B, B′_{0} ∩ N must be conjugate by a unitary element in N. Q.E.D.

4.2. Definition. Let M be a type II_{1} factor. An HT m.a.s.a. of M is a maximal abelian ∗-subalgebra A ⊂ M satisfying the conditions:

1^{∘}. The normalizer of A in M acts ergodically on A, i.e., A is a semiregular m.a.s.a. in M.

2^{∘}. M has the property H relative to A.

3^{∘}. A has a von Neumann subalgebra A_{0} ⊂ A such that A′_{0} ∩ M = A and such that A_{0} ⊂ M is a rigid embedding (equivalently, the pair (M, A_{0}) has the property T).

If a type II_{1} factor M has a HT m.a.s.a then we say that M is a HT factor. We denote by HT the class of HT factors. If M ∈ HT and A ⊂ M is a HT m.a.s.a. of M then we say that A ⊂ M is a HT inclusion, or that (M, A) is a HT pair.

We will often consider the case when A is a Cartan subalgebra of N (i.e., when the normalizer of A in N generates all N, N (A)′′ = N), in which case we call A a HT Cartan subalgebra.

The next 2 Corollaries are immediate consequences of the Theorem 4.1:

4.3. Corollary. 1^{∘}. Any two HT m.a.s.a.’s of a type II_{1} factor are conjugate by a unitary element of that factor.

2^{∘}. If (A_{1} ⊂ M_{1}), (A_{2} ⊂ M_{2}) are HT inclusions and θ is an isomorphism from M_{1} onto M_{2} then there exists a unitary element u ∈ M_{2} such that uθ(A_{1})u^{∗} = A_{2}.

4.4. Corollary. Let A ⊂ M be a HT inclusion. Any automorphism of M can be perturbed by an inner automorphism to an automorphism that leaves A invariant, i.e.,

AutM/IntM = Aut(M, A)/Int(M, A).

The next two results give some basic properties of the HT inclusions, and thus of the HT class of factors.

4.5. Theorem. 1^{∘}. If M ∈ HT and t > 0 then M^{t} ∈ HT .

2^{∘}. If M_{1}, M_{2} ∈ HT then M_{1} ⊗M_{2} ∈ HT . More precisely, if (A_{i} ⊂ M_{i}), i = 1, 2, are HT inclusions then (A_{1} ⊗ A_{2} ⊂ M_{1} ⊗ M_{2}) is a HT inclusion.

Proof. 1^{∘} By 2.6 (respectively 3.7) it follows that if A ⊂ M has the relative property H (resp. is a rigid embedding) and D denotes the diagonal of M_{0} = M_{n×n}(ℂ) then A ⊗ D ⊂ M_{n}(M) has the property H (resp., is rigid). Then by 2.6 (resp. 3.7) again, if we cut this latter inclusion by a projection in A ⊗ D of trace τ(p) = t/n then we get that M^{t} has both property H and T with respect to an appropriate “amplification” of A by t. Thus, M^{t} is a HT factor.

2^{∘}. This follows by applying 2.6 and 3.7 again. Q.E.D.

4.6. Theorem. 1^{∘}. Let N ⊂ M be an inclusion of type II_{1} factors of finite Jones index. Then N is HT if and only if M is HT.

2^{∘}. Let N ⊂ M be an irreducible inclusion of finite index. Assume both N, M are HT factors having HT Cartan subalgebras. Then [M : N] ∈ ℕ.

Proof. 1^{∘}. By taking into account that the basic construction N ⊂ M ⊂ M_{1} has the property that M_{1} is an amplification of N, it follows that it is sufficient to prove that if N ∈ HT then M ∈ HT . Let A ⊂ N be a HT m.a.s.a. and A_{0} ⊂ A be a von Neumann subalgebra such that A_{0} ⊂ N is a rigid embedding and A′_{0} ∩ N = A. Then the inclusion A ⊂ A′ ∩ M has index ≤ [M : N] by commuting squares and it also has a finite orthonormal basis (see e.g., [Po7]). Thus, by Sections 2 and 3 it follows that A′ ∩ M ⊂ M has the relative property H and is rigid. Moreover, by (Section 1 in [Po7]), there exists a projection p ∈ A′∩M such that Ap = p(A′∩M)p. Thus, Ap follows a m.a.s.a. in pMp. Let us show that it is semiregular in pMp.

Let f ∈ A be the support of p in A. Note that Af is semiregular in fNf. For each u ∈ N (Af) let σ_{u} be the automorphism of Af implemented by Adu. It follows that sigma_{u}(a)u = ua, ∀a ∈ Af, so that σ_{u}(a)pup = pupa, ∀a ∈ Af. It follows that the (pu^{∗}pup)^{1/2} ∈ Ap (because Ap is maximal abelian) and that the partial isometry u_{p} in the polar decomposition of pup is in the normalizing groupoid G_{p} of Ap in pMp. Thus, (Ap ∪ G_{p})′′ contains the linear p(spN)p. Thus, if there exists a projection q ∈ (N_{pMp}(Ap))′ ∩ pMp then q commutes with p(spN)p.

Let N_{0} denote the von Neumann algebra generated the normalizer of A in N and N_{1} denote the von Neumann algebra generated by p and N_{0} in M. Since the ∗-algebra spN (A) is weakly dense in N_{0} it follows from the above that the von Neumann algebra generated by Ap and G_{p} contains pN_{0}p and the von Neumann algebra it generates, thus it contains pN_{1}p, thus it is equal to pN_{1}p.

It follows that q commutes with pN_{1}p. By (1.2 in [Po]) it follows that there exists a projection q_{1} in N_{1}′ ∩ M such that q = q_{1}p. But N_{1}′ ∩ M is contained in N_{0}′∩M, which is a finite dimensional von Neumann algebra (because any projection in N_{0}′ ∩ M expects on N as a scalar). Let q_{0} be a minimal projection in N_{1}′ ∩ M under q_{1} and denote e = q_{0}p. Then e ∈ Ap (the latter being maximal abelian in pMp) and the normalizing groupoid of Ae in eMe generates a von Neumann algebra equal to eN_{1}e, which since q_{0} was minimal in N_{1}′ ∩ M follows a factor.

Altogether, this shows that Ae is a semiregular m.a.s.a. in eMe. Also, by (Section 1 in [Po7]) it follows that there exists a finite dimensional subalgebra A_{1} is Ae such that if we denote A_{01} = A_{0}e ∨ A_{1} then A′_{01} ∩ eMe ⊂ Ae. But by Section 3, A_{0}e is rigid in eMe, and by applying again Section 3, it follows that A_{01} is rigid in eMe as well.

2^{∘}. Let A ⊂ N be a HT Cartan subalgebra of N. Since the normalizer N_{N}(A) leaves A′ ∩M invariant and N_{N}(A)′ ∩M = ℂ, it follows by (Section 1 in [Po7]) that the center of A′ ∩ M has uniform number of points above each point of A and that A′ ∩ M is homogeneous of type I_{n}. Thus, there exists a finite orthonormal basis {m_{j}}_{j} of A′ ∩ M over A′ ∩ N = A such that E_{A}(m_{j}^{∗}m_{j}) = 1, ∀j. By commuting squares, E_{N}(m_{j}^{∗}m_{j}) = 1 as well. Thus, N has integer index in the factor N_{0} = N ∨ A′ ∩ M.

Moreover, like in the proof of part 1^{∘}, there exists a projection e ∈ A′ ∩ M such that Ae is maximal abelian in eMe and Ae is a semiregular HT m.a.s.a. in eMe. By Theorem 4.1, it follows that Ae is conjugate to the HT Cartan subalgebra of eMe (which exists by hypothesis). Thus, eN_{o}e must have integer index in eMe because it contains a Cartan subalgebra of eMe (cf. [Po8]). Q.E.D.

4.7. Corollary. If M ∈ HT has HT Cartan subalgebras and N ⊂ M is a subfactor with finite Jones index then:

1^{∘}. [M : N] ≤ 4 implies [M : N] ∈ {1, 2, 3, 4}.

2^{∘}. For each irreducible M-bimodule H appearing in some L^{2}(M_{n}), n ≥ 0, we have dim(_{M}H_{M}) ∈ ℕ.

5. Exemples of HT factors.

In this section we provide several examples of HT factors. All these factors have HT Cartan subalgebras, being obtained via the group measure space construction using cross products by certain special groups. We begin with the simplest example:

5.1. Exemple. Let (X_{0}, μ_{0}) be the 2-dimensional thorus, regarded as the dual group of ℤ^{2}, endowed with the Haar measure μ_{0}. Note that μ_{0} is the same as the Lebesgue measure on X_{0} = 𝕋^{2}. Let σ_{0} be the action of SL(2, ℤ) on X_{0} implemented by the action of SL(2, ℤ) on ℤ^{2}. Let (X_{1}, μ_{1}) be a probability space with a measure preserving ergodic transformation σ_{1} of SL(2, ℤ) on it. Let σ = σ_{0} × σ_{1} be the product action on the probability space (X, μ) = (X_{0} × X_{1}, μ_{0} × μ_{1}). Denote A = L^{∞}(X, μ) and M = A ⋊_{σ} SL(2, ℤ). Also, denote A_{0} = L^{∞}(X_{0}, μ_{0}) and M_{0} = A_{0} ⋊_{σ_{0}} SL(2, ℤ).

Note that we can regard A_{0} as a subalgebra A, in which case the canonical unitaries u_{g}, g ∈ SL(2, ℤ) ⊂ M implementing the action σ on A also implement the action σ_{0} on A_{0}. Thus, M_{0} can be viewed as a subfactor of M. Moreover, we can view A_{0} as L(ℤ^{2}), in which case M_{0} is identified with L(ℤ^{2} ⋊ SL(2, ℤ)).

Given any arbitrary ergodic action σ_{1} as in 5.1, the action σ and the algebras defined above have the following properties:

5.2. Proposition. 1^{∘}. The action σ is a free, ergodic action of SL(2, ℤ) on the probability space (X, μ).

2^{∘}. The action σ_{0} is strongly ergodic. The action σ = σ_{0} × σ_{1} is strongly ergodic if and only if σ_{1} is strongly ergodic.

3^{∘}. There exist ergodic actions σ_{1} of SL(2, ℤ) that are not strongly ergodic. 4^{∘}. M_{0} is a non-Γ type II_{1} factor and A_{0} ⊂ M_{0} is a Cartan subalgebra of M_{0}.

More generally, M is a type II_{1} factor, A ⊂ M is a Cartan subalgebra in M and M is non-Γ if and only if σ_{1} is strongly ergodic.

5^{∘}. When regarded as a subalgebra in M, A_{0} satisfies A′_{0} ∩ M = A.

6^{∘}. A_{0} is rigid in M and M has the property H relative to A. Thus, M is a HT factor and A ⊂ M is a HT Cartan subalgebra in M.

Proof. 1^{∘}. It is well known that a non-inner automorphism of a group G implements a properly outer automorphism on L(G) that preserves the canonical trace of L(G). Thus, since each non-trivial element g in the group SL(2, ℤ) implements a non-inner automorphism of ℤ^{2} it follows that σ_{0}(g) is properly outer ∀g ∈ SL(2, ℤ), g ≠ e.

Since the tensor product of any properly outer automorphism with an arbitrary automorphism is properly outer, it follows that σ(g) is properly outer ∀g ∈ SL(2, ℤ), g ≠ e, as well.

Furthermore, the action σ_{0} is well known to be mixing. More precisely, it is easy to see that given any finite set F ⊂ SL(2, ℤ), e ∉ F, there exists g ∈ SL(2, ℤ) such that gF ∩ F = ∅. But this implies not only that σ_{0} is ergodic, but also that its tensor product with any ergodic action is still ergodic.

2^{∘}. The first part is a well known result of Klaus Schmidt ([S1]). The second part is an immediate consequence of the proof of this result in ([S1]).

3^{∘}. This is a consequence of a theorem of Connes and Weiss ([CW]), showing that any discrete group G_{1} which doesn’t have the property T has a free, ergodic but not strongly ergodic action σ_{1} on a probability space. Thus, one simply applies this result to G_{1} = SL(2, ℤ), which doesn’t have the property T.

One can in fact avoid using the general result in ([CW]), by noticing that since SL(2, ℤ) has an infinite amenable group H as a quotient (see e.g., [dHV]), any ergodic action of H on a non-atomic probability space (e.g., a Bernoulli shift action of H) composed with the quotient map G_{1} → H gives an ergodic but not strongly ergodic action of SL(2, ℤ) (note that the resulting action of SL(2, ℤ) is not free though, in fact not even faithful, but freeness is not necessary in the construction 1.1).

4^{∘}. Since SL(2, ℤ) is close to be a free group (see e.g., [dHV]), it is easy to see that any central sequences in a type II_{1} factor of the form B ⋊_{σ} SL(2, ℤ) obtained as the cross product of a finite von Neumann algebra (B, τ) by a free, τ-preserving action σ of SL(2, ℤ) on it must be supported on B.

5^{∘}. Note that if b = Σ_{g}a_{g}u_{g} ∈ M = A ⋊_{σ} SL(2, ℤ) commutes with all a ∈ A_{0} and a_{g} ≠ 0 for some g ≠ e then aa_{g}u_{g} = a_{g}u_{g}a, ∀a ∈ A_{0}, implying that a_{g}a = a_{g}σ_{g}(a), ∀a ∈ A_{0}. This in turn contradicts the fact that σ_{g} is properly outer on A_{0}. Thus, all a_{g} with g ≠ 0 must be equal to 0, implying that b lies in A.

6^{∘}. By ([Kaz]), ℤ^{2} ⊂ ℤ^{2} ⋊ SL(2, ℤ) is a rigid inclusion of groups. Equivalently, A_{0} ⊂ M_{0} is rigid, so even more so A_{0} ⊂ M is rigid. Moreover, since A ⊂ M is a cross-product inclusion by an action of SL(2, ℤ), which has Haagerup compact approximation property, it follows that M has the property H relative to A. Q.E.D.

5.3. Corollary. Let M be any of the factors constructed in 5.1, with its corresponding HT Cartan subalgebra A ⊂ M. For each n ≥ 2 let p_{n} ∈ A be a projection of trace τ(p_{n}) = 1/12(n − 1). Then (Ap_{n} ⊂ p_{n}Mp_{n}) is isomorphic to (L^{∞}([0, 1]) ⊂ L^{∞}([0, 1]) ⋊ 𝔽_{n}), for some free ergodic action of 𝔽_{n} on L^{∞}([0, 1]), and Ap_{n} ⊂ p_{n}Mp_{n} are HT inclusions. The corresponding HT factors M_{n} = L^{∞}([0, 1]) ⋊ 𝔽_{n}, n ≥ 2, are stably isomorphic, namely, if n > m and q_{nm} ∈ M_{m} satisfies τ(q_{nm}) = (m − 1)/(n − 1) then q_{nm}M_{m}q_{nm} ≃ M_{n}.

Proof. The first part is a consequence of a recent result of G. Hjorth ([Hj]). The second part is then an immediate consequence of 2.5 and 3.8. Q.E.D.

The second class of examples is based on the result of A. Valette ([Va]), mentioned in 3.8, and on ([dCaH], [CowH]), mentioned in 2.5:

5.4. Proposition. For any n ≥ 2 there exists a lattice Γ in SO(n, 1) (respectively in SU(n, 1)) for which there exist free, ergodic, measure-preserving actions σ on the standard probability space (X, μ) such that A = L^{∞}(X, μ) is a HT Cartan subalgebra in M = A ⋊_{σ} Γ.

Our final set of examples shows that for some special classes of HT factors (the ones obtained by group measure space construction), taking cross products by certain actions of property H groups, keeps us in the HT class.

5.5. Proposition. Let A_{0} ⊂ A_{0} ⋊_{σ_{0}} G_{0} be a HT inclusion and G_{1} a discrete group with the property H. Let

A = \overline{⊗_{h∈G_{1}}}(A_{0}, τ_{|A_{0}})_{h}

and still denote by σ_{0} the action of G_{0} on this infinite tensor product given by σ_{0}(g)(⊗_{h}a_{h}) = ⊗_{h}σ_{0}(g)(a_{h}), ∀g ∈ G_{0}. Denote by σ_{1} the Bernoulli shift action of G_{1} on A Noting that σ_{0} commutes with σ_{1}, let σ denote the product action σ_{0} × σ_{1} of G = G_{0} × G_{1} on A. Then A ⊂ A ⋊_{σ} G is a HT inclusion.

Proof. Since G_{0}, G_{1} have the property H, it follows that G = G_{0} × G_{1} has the property H. Also, if A_{00} ⊂ A_{0} is so that A′_{00} ∩ M_{0} = A_{0} and A_{0} ⊂ M_{0} is rigid, where M_{0} = A_{0} ⋊_{σ_{0}} G_{0}, then by the construction of A and σ_{1}, we clearly have A′_{00} ∩ M = A, where M = A ⋊_{σ} G. Q.E.D.

6. Betti numbers for HT factors and applications.

6.1. Definition. Let M be a HT factor and A ⊂ M a HT m.a.s.a. of M. Let R_{M} be the countable measurable orbit equivalence relation (called standard equivalence relation in [G1,2]) implemented on A by N (A) (cf. [FM]). Let {β_{n}(R_{M})}_{n≥0} be the ℓ^{2} Betti numbers of R_{M}, as defined by D. Gaboriau in ([G2]). We denote β_{n}(M) =^{def} β_{n}(R_{M}), n = 0, 1, 2, ... and call them the Betti numbers of M. Note right away that by ([G2]), β_{0}(M) is always equal to zero.

From the results in Section 4 and the properties proved by Gaboriau for the Betti numbers of the standard equivalence relations, one immediately gets:

6.2. Corollary. . 1^{∘}. If A ⊂ A ⋊_{σ} G is a HT inclusion, for some countable discrete group G acting freely and ergodically on A ≃ L^{∞}(X, μ), then β_{n}(M) is equal to the n’th ℓ^{2} Betti number of G, as defined in [ChG]).

2^{∘}. If M ∈ HT and t > 0 then β(M^{t}) = β(M)/t, ∀n.

3^{∘}. If M_{1}, M_{2} ∈ HT then for each n ≥ 0 we have:

β_{n}(M_{1} ⊗ M_{2}) = Σ_{i+j=n}β_{i}(M_{1})β_{j}(M_{2})

Proof. 1^{∘}. By 6.1, we have β_{n}(M) = β_{n}(R_{M}). But R_{M} = R_{G}, and by Gaboraiu’s theorem the latter has Betti numbers β(R_{G}) equal to the Cheeger-Gromov ℓ^{2} Betti numbers β_{n}(G) of the group G.

2^{∘}. By Section 4 we know that the class HT is closed to amplifications and tensor products. Moreover, it is immedaiate to see that the “amplification” of a semiregular m.a.s.a. A ⊂ M has a normalizer that gives rise to the standard equivalence relation (R_{M})^{t}, as defined in [G2]. Then formula 2^{∘} is a consequence of Gaboriau’s similar result for standard equivalence relations.

Part 3^{∘} follows similarily, by taking into account that if A_{1} ⊂ M_{1}, A_{2} ⊂ M_{2} are semiregular m.a.s.a.’s then N (A_{1} ⊗ A_{2})′′ = (N (A_{1}) ⊗ N (A_{2}))′′ (this is a trivial consequence of 1.2 in [JPo], for instance). Q.E.D.

6.3. Corollary. 1^{∘}. If M ∈ HT has at least one non-zero Betti number, then it has trivial fundamental group, F(M) = {1}.

2^{∘}. If M ∈ HT has at least one non-zero Betti number then M^{t_{1}} ⊗ ... ⊗ M^{t_{n}} is isomorphic to M^{s_{1}} ⊗ ... ⊗ M^{s_{m}} if and only if n = m and t_{1}...t_{n} = s_{1}...s_{m}.

Proof. Part 1^{∘} is trivial by the formula β_{n}(M^{t}) = β(M)/t. Part 2^{∘} follows from the fact that if β_{n_{i}}(M_{i}) is the first non-zero Betti number for M_{i}, i = 1, 2, (which thus necessarily satisfies n_{i} ≥ 1), then β_{n}(M_{1} ⊗ M_{2}) = 0 if n < n_{1} + n_{2}. Q.E.D.

6.4. Corollary. The factors M constructed in 5.1 have the following properties:

1^{∘}. For each n ≥ 2 let p_{n} ∈ A be a projection of trace τ(p_{n}) = 1/12(n−1). Then (Ap_{n} ⊂ p_{n}Mp_{n}) ≃ (L^{∞}([0, 1]) ⊂ L^{∞}([0, 1]) ⋊ 𝔽_{n}), for some free ergodic action of 𝔽_{n} on L^{∞}([0, 1]). The corresponding factors M_{n} = L^{∞}([0, 1]) ⋊ 𝔽_{n}, n ≥ 2, are mutually non-isomorphic, but stably isomorphic, namely, if n > m and q_{nm} ∈ M_{m} satisfies τ(q_{nm}) = (m − 1)/(n − 1) then q_{nm}M_{m}q_{nm} ≃ M_{n}.

2^{∘}. Two tensor products M^{t_{1}} ⊗ ... ⊗ M^{t_{n}} and M^{s_{1}} ⊗ ... ⊗ M^{s_{m}} are isomorphic if and only if n = m and t_{1}t_{2}...t_{n} = s_{1}s_{2}...s_{m}. In particular, F(M) = {1}.

3^{∘}. All subfactors of finite index of M, as well as of the tensor products in 2^{∘} (in particular, of the factors M_{n} in 1^{∘}), are extremal and have integer index. Moreover, given any n ∈ ℕ there exist irreducible subfactors N ⊂ M, N_{0} ⊂ M such that [M : N] = [M : N_{0}] = m and such that β_{n}(N) = mβ_{n}(M) while β_{n}(N_{0}) = β_{n}(M), for all n ≥ 0.

4^{∘}. The factor M cannot be embedded into a free group factor. Also, M does not contain subfactors L(G) with G an infinite property T group.

5^{∘}. The action σ_{1} in the construction of M can be taken either strongly ergodic (i.e., without asymptotically invariant sequences), in which case M follows non-Γ, or non - strongly ergodic, in which case M has the property Γ. Thus, there exist property Γ factors with trivial fundamental group.

Proof. 1^{∘}, 2^{∘}, the first part of 3^{∘} and 5^{∘} have already been proved.

For the last part of 3^{∘}, to construct N just take a subgroup of index m in 𝔽_{fin}, which exists for any m ∈ ℕ. To obtain N_{0} take the action σ_{1} on (X_{1}, μ_{1}) so that to leave invariant a quotient (Y_{1}, λ_{1}) of X that has exactly m points in each fiber.

The first part of 4^{∘} has the same proof as the similar result in ([CJ1]), as for instance done in ([Po1]). The second part in 4^{∘} has a similar proof to the proof of 4.1 (except easier, in terms of proving a lemma playing the role of 1.1). Q.E.D.

The examples 5.4, 5.5 give additional, distinct families of type II_{1} factors with trivial fundamental group:

6.5. Corollary. 1^{∘}. Let A, Γ, σ be like in Proposition 5.4, with Γ a lattice in SO(2n, 1) or SU(n, 1). Let M_{n} = A ⋊_{σ} Γ. Then F(M_{n}) = {1}, and in fact each M_{n} satisfies the same property as in 6.4.2^{∘}, ∀n. Moreover, the factors M_{n} are mutually non-stably isomorphic, for different n’s.

2^{∘}. Given any group G_{0} which is either tree-able in the sense of ([Ga1]) (e.g. G_{0} = SL(2, ℤ) or G_{0} = 𝔽_{n} for some n ≥ 2), or is equal to some lattice Γ like in 5.4, and any group G_{1} with the property H, there exists a HT factor M with β_{n}(M) = β_{n}(G_{0} × G_{1}), ∀n ≥ 0.

Proof. By 6.2 and ([G2], [ChG]), each M_{n} has exactly one non-zero beta number β_{k_{n}}(M_{n}) ≠ 0 and k_{n} ≠ k_{m} if n ≠ m. This proves 1^{∘}.

Part 2^{∘} is a consequence of 5.5. Q.E.D. 6.6. Remarks. 1^{∘}. Note that the above Corollaries 6.4 and 6.5 solve Problem

3 from Kadison’s Baton Rouge list, providing lots of examples of factors M with the property that the algebra of n by n matrices over M is not isomorphic to M, for any n ≥ 2.

2^{∘}. In a recent conference at MSRI ([C6]). Connes addressed the question of contsructiong Betti number invariants for typer II_{1} factors, building on the same conceptual grounds as in the case of measurable foliations ([C4]), dicrete groups ([ChGr]), and ultimately equivalence relations ([G2, 3]). Although our results do not quite address the question this way (see remarks 3^{∘} − 6^{∘} below), they bring some light to the problem, pointing out a significant phenomenology which we hope could serve both to obtain further results and as a test case for better definitions of Betti numbers.

3^{∘}. It should be noted that, although the notion Betti numbers introduced here is for global factors M, it is in some sense a “relative” notion, in that it only applies for factors having certain special m.a.s.a.’s A ⊂ M and it reduces to the notion of Betti numbers for A ⊂ M, modulo proving a uniqueness result for such m.a.s.a.’s. We avoided to complicate the notations and opted for “β_{n}(M)”, thus not indicating the “relative” aspect of this notion, mostly for convenience of notations.

Thus, if it was to indicate “r” for “relative”, the fact that we take semiregular m.a.s.a. as the subalgebra “relative to which” we built the theory should also have been indicated (thus making the notation quite complicated).

This is because in exactly the same spirit as the notion introduced here based on HT Cartan subalgebras, one can consider the class of factors of the form M = R ⋊_{σ} G_{0}, with R the hyperfinite type II_{1} factor, G_{0} a group with the property H and the actions σ chosen so that R ⊂ M be a rigid inclusion, as defined in Section 3. Such actions exist for G = SL(2, ℤ), by ([Cho]), so the class is non-empty. A result similar to Theorem 1.1 can then be proved for the “HT hyperfinite subfactors” R and the same proof as for 4.1 shows that a factor M contains a unique such subfactor R, modulo unitary conjugacy.

4^{∘}. The same strategy can of course be played with other invariants than the Betti numbers considered here, and possibly with respect to other type of subalgebras than m.a.s.a.’s (see also the Remark 3^{∘} above). In a forthcoming paper, we will develop an invariant for type II_{1} factors M obtained from cross products of the hyperfinite type II_{1} factor R by Connes-Stormer Bernoulli shift actions of property T groups G, and use the invariants introduced in ([Po9]) for such inclusions R ⊂ M.

5^{∘}. It would be very important to extend the class of factors in the “good class” for which a certain uniqueness result can be proved for some special type of Cartan subalgebras, beyond the HT factors we considered here. Such generalizations can go in many directions: by either extending the class of groups G_{0} for which A ⊂ A ⋊_{σ} G_{0} works, for certain σ, or by showing that for the groups G_{0} we already considered here (e.g., the free groups) any action σ works.

6^{∘}. There is of course an analogy between factors of the form A ⋊ 𝔽_{n} and the free group factors L(𝔽_{n}) that gives room to speculations about the possibility that, like their cross-product counter-parts, the free group factors are non-isomorphic, for different n’s. In this respect, there are already many indications from results in Voiculescu’s free probability that this would indeed be the case (cf. [V1,2], [Ra], [Dy], [Ge], [Sh], etc.). Note however that such free group factors have no Cartan subalgebras ([V2]), and that if some formula for Betti numbers is to hold true for group von Neumann algebras, as in Connes’ problem ([C6]), then according to Voiculescu’s formula ([V1]) for the number of generators of the amplifications/compressions of the free group factors (cf. also [Ra], [Dy]), the Betti number β_{1} (= number of generators −1) should satisfy a formula of the type β(M^{t}) = β(M)/t^{2} !

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