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Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of \(w\)-rigid groups. I. (English) Zbl 1120.46043
The paper under review gives a remarkable answer to a challenging problem concerning type II\(_1\) factors, originating in the classical work of F. J. Murray and J. von Neumann [Ann. Math. 44, 716–808 (1943; Zbl 0060.26903)]. For any subgroup \(S\subset {\mathbb R}_+^*\) the existence of a II\(_1\) factor \(M\) with fundamental group \({\mathcal F}(M)=S\) is proved. When \(S\) is countable \(M\) can be chosen to be a separable factor (i.e., \(M_*\) is separable). Previous progress was made in 1943 by Murray and von Neumann (proving \({\mathcal F}(R)={\mathbb R}_+^*\) for the hyperfinite II\(_1\) factor \(R\)), in 1980 by A. Connes [J. Oper. Theory 4, 151–153 (1980; Zbl 0455.46056)] (proving that \({\mathcal F}(M)\) is countable if \(M\) is the von Neumann algebra of a discrete countable ICC group with Kazhdan’s property T), in 1987 by V. Ya. Golodets and N. I. Nessonov [J. Funct. Anal. 70, 80–89 (1987; Zbl 0614.46053)] (constructing, for each countable set \(\Sigma\subset {\mathbb R}_+^*\), a type II\(_1\) factor \(M\) with \({\mathcal F}(M)\) countable and \(\Sigma\subset {\mathcal F}(M)\)), and very recently by the author [Ann. Math. (2) 163, 809–899 (2006; Zbl 1120.46045)] (for any II\(_1\) factor \(M\) with a Cartan subalgebra satisfying simultaneously a relative Haagerup type compact approximation property and a relative property T, e.g., \(M={\mathcal L}({\mathbb Z^2} \rtimes SL_2({\mathbb Z}))\), one has \({\mathcal F}(M)=\{ 1\}\)).
The class of II\(_1\) factors considered here consist of crossed products \(M=N \rtimes_\sigma G\) with two properties. The first one requires the discrete ICC group \(G\) to be { \(w\)-rigid} (i.e., to contain an infinite normal subgroup with the relative property T). The second one requires the action \(\sigma\) to be { malleable} and { mixing}. Malleability, as defined by the author [J. Funct. Anal. 230, 273–328 (2006; Zbl 1097.46045)], is related with realizability of the finite von Neumann algebra \((N,\tau)\) as the centralizer \({\mathcal N}_\varphi\) of a von Neumann algebra with discrete decomposition \(({\mathcal N},\varphi)\) and with the existence of a { gauged extension} \(\tilde{\sigma}\) of \(\sigma\) to \((\tilde{{\mathcal N}},\tilde{\varphi})=({\mathcal N}\otimes {\mathcal N},\varphi\otimes \varphi)\) which leaves \({\mathcal N}={\mathcal N}\otimes 1\subset \tilde{{\mathcal N}}\) invariant and commutes with a certain action \(\alpha\) of \({\mathbb R}\) on \(\tilde{{\mathcal N}}\) for which \(\alpha_1 ({\mathcal N}\otimes 1)=1\otimes {\mathcal N}\). The almost periodic spectrum \(S({\mathcal N},\varphi)\), given by the pure point spectrum of the modular group \(\Delta_\varphi\), is a subgroup \(S(\tilde{\sigma})\subset {\mathcal F} (N\rtimes_\sigma G)\subset {\mathbb R}_+^*\). Moreover, the inclusion \(M=N\rtimes_\sigma G \subset {\mathcal N}\rtimes_{\tilde{\sigma}} G\) generates a family \(\text{Aut}_t (M;\tilde{\sigma})\) of \(t\)-scaling automorphisms, \(t\in S(\tilde{\sigma})\), such that any two of them differ by an inner automorphism of \(M\). The embeddings of \({\mathcal L}(G)\) in \(N\rtimes_\sigma G\) are very rigid, as shown by the following central result in the paper.
Theorem. Let \(G_i\) be \(w\)-rigid ICC groups, \(\sigma_i :G_i \rightarrow \text{ Aut} (N_i,\tau_i)\) malleable mixing actions with gauged extensions \(\tilde{\sigma}_i\) and \(M_i=N_i \rtimes_{\sigma_i} G_i\), \(i=0,1\). Let \(\theta:M_0 \cong M_1^s\) be an isomorphism for some \(s>0\). Then there exist unique \(\beta_i \in S(\tilde{\sigma}_i)\) and unique \(\theta^i_{\beta_i} \in \text{ Aut}_{\beta_i} (M_i; \tilde{\sigma}_i)\) such that \(\theta^1_{\beta_1} (\theta ({\mathcal L}(G_0)))={\mathcal L}(G_1)^{s\beta_1}\) and \(\theta (\theta^0_{\beta_0} ({\mathcal L}(G_0)))={\mathcal L}(G_1)^{s\beta_0}\). Moreover, \(\beta_0=\beta_1\).
In particular, taking \(G_i=G\), \(\sigma_i=\sigma\), \(M_i=N\rtimes _\sigma G\) with \(G\) a \(w\)-rigid group and \(\sigma\) a malleable mixing action, one has
\[ \forall s\in {\mathcal F}(N\rtimes _\sigma G),\;\exists t\in S(\tilde{\sigma}),\;st\in {\mathcal F}({\mathcal L} (G)). \]
Corollary. Let \(G\) be a discrete ICC \(w\)-rigid group for which \({\mathcal F}({\mathcal L}(G))=\{ 1\}\), for instance \(G={\mathbb Z}^2 \rtimes SL_2({\mathbb Z})\), and \(\sigma\) be a malleable mixing action on \(N\). Then \({\mathcal F}(N\rtimes _\sigma G)=S(\tilde{\sigma})\).
Bernoulli \(G\)-shifts (both of classical and Connes-Størmer type) and Bogoliubov shifts provide prominent examples of malleable mixing actions. Classical (commutative) Bernoulli \(G\)-shifts \(\sigma\) on \(\bigotimes_{g\in G} {\mathbb T}\) only give \(S(\tilde{\sigma})=\{ 1\}\) and \(\text{Aut}(M;\tilde{\sigma})=\text{ Inn} (M)\). Myriads of examples are however produced by Connes-Størmer \(G\)-shifts \(\sigma\) on ITPFI factors \(({\mathcal N},\varphi)=\bigotimes_{g\in G} (B({\mathcal H}),\varphi_0)\). When the state \(\varphi_0\) is defined by the list of eigenvalues \(\{ t_i \}_i\), the group \({\mathcal F}(N\rtimes _\sigma G)=S(\tilde{\sigma})\) coincides with the multiplicative subgroup in \({\mathbb R}_+^*\) generated by the ratios \(\{ t_i / t_j\}_{i,j}\); thus any subgroup of \({\mathbb R}_+^*\) is effectively realized as the fundamental group of a type II\(_1\) factor.

46L35 Classifications of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
22F10 Measurable group actions
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
46L10 General theory of von Neumann algebras
46L40 Automorphisms of selfadjoint operator algebras
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