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On a class of type $$\text{II}_1$$ factors with Betti numbers invariants. (English) Zbl 1120.46045
The fundamental work of F. J. Murray and J. von Neumann [Ann. Math. 44, 716–808 (1943; Zbl 0060.26903)] associates to every type II$$_1$$ factor $$M$$ the subgroup $${\mathcal F} (M)=\{ t>0: M^t \cong M\}$$ of the multiplicative group $${\mathbb R}_+^*$$, called the fundamental group of $$M$$. The uniqueness of the hyperfinite type II$$_1$$ factor $$R$$ immediately gives $${\mathcal F}(R)={\mathbb R}_+^*$$. However, Murray and von Neumann noticed (last line of Chapter V) that they know nothing beyond this example. The related problem of the existence of a type II$$_1$$ factor $$M$$ with $$M\ncong M \otimes M_n ({\mathbb C})$$, $$n\geq 2$$, was raised by Kadison (Baton Rouge Conference, 1967) and by [S. Sakai, $$C*$$-Algebras and $$W^*$$-Algebras, Springer (1998; Zbl 1024.46001)]. A first class of examples of factors with countable fundamental group was obtained by A. Connes [J. Oper. Theory 4, 151–153 (1980; Zbl 0455.46056 ]. He proved that $${\mathcal F}(M)$$ is countable if $$M={\mathcal L}(\Gamma)$$, the von Neumann algebra of a discrete countable group $$\Gamma$$ with infinite conjugacy classes and with Kazhdan’s property (T). Using noncommutative Bernoulli shifts, V. Y. Golodets and N. I. Nessonov [J. Funct. Anal. 70, 80–89 (1987; Zbl 0614.46053)] constructed, for each countable set $$S\subset {\mathbb R}_+^*$$, a type II$$_1$$ factor $$M$$ with $${\mathcal F}(M)$$ countable and $$S\subset {\mathcal F}(M)$$.
The paper under review exhibits the first class of type II$$_1$$ factors with explicitly computable countable fundamental group. This includes the von Neumann algebra $$M={\mathcal L}(\Gamma)$$ of the group $$\Gamma={\mathbb Z}^2 \rtimes SL_2 ({\mathbb Z})$$, for which the equality $${\mathcal F}(M)=\{ 1\}$$ is proved. In particular this $$M$$ cannot be isomorphic to any of the factors $$M\otimes M_n({\mathbb C})$$, providing an answer to the Kadison-Sakai question.
The key idea is to prove that D. Gaboriau’s $$\ell^2$$-Betti numbers $$\{ \beta_n ({\mathcal R})\}_{n\geq 0}$$ associated to a countable measure-preserving equivalence relation $${\mathcal R}$$ [Publ. Math., Inst. Hautes Étud. Sci. 95, 93–150 (2002; Zbl 1022.37002)] produce numeric invariants for a certain class $${\mathcal H}{\mathcal T}$$ of type II$$_1$$ factors. This class consists of factors $$N$$ possessing a Cartan subalgebra $$B$$ with property HT, i.e., satisfying a certain weak amenability type property (the inclusion $$B\subset N$$ satisfies a Haagerup type compact approximation property) and a weak rigidity property (there exists $$B_0 \subset B$$ subalgebra s.t. $$B_0^\prime \cap N\subset B$$ and the embedding $$B_0 \subset B$$ is a property T – or rigid – inclusion, as defined by the author). The central technical result in the paper is that any two HT Cartan subalgebras in a type II$$_1$$ factor $$M\in {\mathcal H}{\mathcal T}$$ are conjugated by a unitary in $$M$$. In conjunction with the Feldman-Moore correspondence between embeddings $$A\simeq L^\infty (X,\mu)\subset M$$ of Cartan subalgebras and pairs of countable ergodic measure-preserving equivalence relations on the probability space $$(X,\mu)$$ and $${\mathcal U}(L^\infty (X,\mu))$$-valued $$2$$-cocycles, this shows that there is a unique such equivalence relation $${\mathcal R}_M^{\text{ HT}}$$ corresponding to $$M$$. Therefore invariants of $${\mathcal R}_M^{\text{ HT}}$$ are also invariants of $$M$$. In particular the Betti numbers $$\beta_n^{\text{ HT}}(M):=\beta_n ({\mathcal R}_M^{\text{ HT}})$$ are invariants of $$M\in {\mathcal H}{\mathcal T}$$. These invariants are well behaved with respect to amplifications and tensor products, operations under which the class $${\mathcal H}{\mathcal T}$$ is shown to be closed, and satisfy $$\beta_n^{\text{ HT}} (M^t)=\beta_n^{\text{ HT}}(M)/t$$, $$n\geq 0$$, $$t>0$$, and a Künneth type formula for the tensor product. As a result the following important corollary is obtained:
Theorem. If $$M\in {\mathcal H}{\mathcal T}$$ has at least one nonzero finite Betti number, then $${\mathcal F}(M)=\{ 1\}$$. In fact $$M^{t_1} \otimes \cdots \otimes M^{t_n} \cong M^{s_1} \otimes \cdots \otimes M^{s_m}$$ if and only if $$n=m$$ and $$t_1\cdots t_n=s_1 \cdots s_m$$. Equivalently, $$\{ M^{\otimes m}\}_{m\geq 1}$$ are stably nonisomorphic and have trivial fundamental group.
When $$M=A\times_\sigma \Gamma_0$$ with $$A$$ an HT Cartan subalgebra and $$\sigma$$ an ergodic free action of a discrete countable group $$\Gamma_0$$ on $$A$$, one has $$\beta_n^{\text{ HT}}(M)=\beta_n (\Gamma_0)$$, the $$n$$th $$\ell^2$$-Betti number of $$\Gamma_0$$ defined by J. Cheeger and M. Gromov [Topology 25, 189–215 (1986; Zbl 0597.57020)] and computed in many important cases (see also [A. Borel, Ann. Acad. Sci. Fenn., 10, 95–105 (1985; Zbl 0586.57022)]), including $$\Gamma_0=SL_2({\mathbb Z})$$ where $$\beta_1=1/12$$ and $$\beta_k=0$$, $$k\neq 1$$. Since $$M_0={\mathcal L}(\mathbb Z^2 \rtimes SL_2(\mathbb Z))\in {\mathcal H}{\mathcal T}$$, one has $${\mathcal F} (M_0)=\{ 1\}$$. In this vein another application is that two factors $$M_\alpha=L_\alpha ({\mathbb Z}^2)\rtimes SL_2({\mathbb Z})$$ and $$M_{\alpha^\prime}$$, with $$SL_2({\mathbb Z})$$ acting naturally on the rotation von Neumann algebra $$L_\alpha ({\mathbb Z}^2)$$ and $$\alpha$$ (resp., $$\alpha^\prime$$) a primitive root of unity of order $$n$$ (resp., $$n^\prime$$), are isomorphic if and only if $$n=n^\prime$$. Many other examples related to free groups and arithmetic lattices in $$SU(n,1)$$ and $$SO(m,1)$$ are discussed. The class $${\mathcal H}{\mathcal T}$$ is also investigated in the framework of subfactors. It is proved to be closed under extensions and restrictions of finite Jones index and to have a highly rigid structure of the lattice of finite index subfactors. The paper concludes with some interesting thoughts on a problem raised in 2001 by Alain Connes concerning the possibility of constructing $$\ell^2$$-Betti number invariants for type II$$_1$$ factors through simplicial complexes and $$\ell^2$$-homology/cohomology for factors.
A direct proof of the equality $${\mathcal F}({\mathcal L} ({\mathbb Z}^2 \rtimes SL_2({\mathbb Z}))=\{ 1\}$$ has appeared in [S. Popa, Proc. Natl. Acad. Sci. USA 101, No.3, 723–726 (2004; Zbl 1064.46048)]. In recent work [Invent. Math. 165, 369–408 (2006; Zbl 1120.46043)] the author has proved the remarkable result that any countable subgroup of $${\mathbb R}_+^*$$ can be realized as the fundamental group of a separable type II$$_1$$ factor.

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 22F10 Measurable group actions 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 46L10 General theory of von Neumann algebras 46L55 Noncommutative dynamical systems
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