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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.

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.

Reviewer: Florin P. Boca (Urbana-Champaign)

### 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 |