Cartan subalgebras of amalgamated free product \(\mathrm{II}_1\) factors. With an appendix by Adrian Ioana and Stefaan Vaes.
(Sous-algèbres de Cartan de produit amalgamé de facteurs de type \(\mathrm{II}_1\).)

*(English. French summary)*Zbl 1351.46058The author studies Cartan subalgebras and, more generally, normalizers of an arbitrary diffuse amenable von Neumann subalgebra of an amalgamated free product of finite von Neumann algebras. The study of Cartan subalgebras is crucial in the classification of group-measure space \(\text{II}_1\) factors: proving that a certain class of group-measure space \(\text{II}_1\) factors \(\text{L}^\infty(X)\rtimes\Gamma\) has a unique Cartan subalgebra reduces their classification, up to isomorphism, to the classification of the corresponding actions, up to orbit equivalence. Both the classification and uniqueness of Cartan subalgebra problems of \(\text{II}_1\) factors coming from actions of amenable groups have been settled since the early 1980’s.

In the last decade, S. Popa’s deformation/rigidity theory has led to spectacular progress in the classification and the uniqueness of Cartan subalgebra problems of \(\text{II}_1\) factors coming from actions of non-amenable groups. The breakthrough work of N. Ozawa and S. Popa [Ann. Math. (2) 172, No. 1, 713–749 (2010; Zbl 1201.46054); Am. J. Math. 132, No. 3, 841–866 (2010; Zbl 1213.46053)] shows how to use the weak amenability of \(\Gamma\) to obtain the uniqueness of the Cartan subalgebra, up to unitary conjugacy, in \(\text{L}^\infty(X)\rtimes\Gamma\) for some profinite actions, and S. Popa and S. Vaes [Acta Math. 212, No. 1, 141–198 (2014; Zbl 1307.46047)] succeeded in removing the profiniteness assumption which allows to give examples of weakly amenable groups \(\Gamma\) which are \(\mathcal{C}\)-rigid, i.e., such that any free and ergodic action gives rise to a \(\text{II}_1\) factor with a unique Cartan subalgebra, up to unitary conjugacy.

One of the major achievement of the paper under review is to get rid of the weak amenability assumption on \(\Gamma\) and still provide uniqueness of Cartan subalgebra results. It is shown that any group \(\Gamma=\Gamma_1*_\Lambda\Gamma_2\) with \([\Gamma_1:\Lambda]\geq 2\), \([\Gamma_2:\Lambda]\geq 3\) and \(\Lambda\) weakly malnormal in \(\Gamma\) is \(\mathcal{C}\)-rigid (Theorem 1.1). In particular, \(\text{SL}_3(\mathbb{Z})*\mathbb{Z}\) is \(\mathcal{C}\)-rigid and non-weakly amenable. Moreover, using S. Popa’s cocycle superrigidity theorem the author concludes that the Bernoulli action of a direct product of two groups as in Theorem 1.1 is \(W^*\)-superrigid (Corollary 1.2).

To show such a result, the author writes the group measure space construction \(\text{L}^\infty(X)\rtimes\Gamma\) with \(\Gamma=\Gamma_1*_\Lambda\Gamma_2\) as a von Neumann algebraic amalgamated free product and studies, in general, the normalizer of an arbitrary diffuse von Neumann subalgebra \(A\) in an amalgamated free product \(M=M_1*_B M_2\) of tracial von Neumann algebras. The more general type of results which can be found in the paper is the following: if \(A\) is amenable relative to \(B\), \(A\nprec_M B\) and \(P'\cap M^\omega=\mathbb C1\), for some free ultrafilter \(\omega\) (where \(P\) is the von Neumann algebra generated by the normalizer of \(A\) in \(M\)), then either \(P\) is amenable relative to \(B\) or \(P\prec_M M_i\) for some \(i\in\{1,2\}\) (Theorem 1.6). The condition \(P'\cap M^\omega\) is not optimal and has been removed in a subsequent work by S. Vaes [Publ. Res. Inst. Math. Sci. 50, No. 4, 695–721 (2014; Zbl 1315.46067)].

Theorem 1.6 has applications in the study of Cartan subalgebras in a general amalgamated free product of tracial von Neumann algebras (Theorem 1.3). In particular, in the context of equivalence relations, it is shown that, for \(\mathcal{R}_1\) and \(\mathcal{R}_2\), two countable ergodic probability measure preserving equivalence relations on a standard probability space \((X,\mu)\) with infinite orbits almost everywhere and without any hyperfinite restrictions, if \(\mathcal{R}=\mathcal{R}_1*\mathcal{R}_2\) is strongly ergodic, then the von Neumann algebra of \(\mathcal{R}\) has a unique Cartan subalgebra (Corollary 1.4). In the context of free products without amalgamation, it is shown that \(M=M_1*M_2\) does not have any Cartan subalgebra whenever \(M_1\) and \(M_2\) are non-trivial tracial von Neumann algebras with \(\text{dim}(M_1)+\text{dim}(M_2)\geq 5\) (Corollary 1.5). In the context of strong solidity, Theorem 1.6 allows to show that the class of strongly solid von Neumann algebras is closed under free products and a large class of amalgamated free products (Theorem 1.8).

The paper contains more general statements and some other results. It also contains an appendix written by A. Ioana and S. Vaes in which S. Popa’s \(w\)-spectral gap property is discussed. It is shown that the technical assumption used in the proof of Theorem 1.6 is actually equivalent to \(w\)-spectral gap.

In the last decade, S. Popa’s deformation/rigidity theory has led to spectacular progress in the classification and the uniqueness of Cartan subalgebra problems of \(\text{II}_1\) factors coming from actions of non-amenable groups. The breakthrough work of N. Ozawa and S. Popa [Ann. Math. (2) 172, No. 1, 713–749 (2010; Zbl 1201.46054); Am. J. Math. 132, No. 3, 841–866 (2010; Zbl 1213.46053)] shows how to use the weak amenability of \(\Gamma\) to obtain the uniqueness of the Cartan subalgebra, up to unitary conjugacy, in \(\text{L}^\infty(X)\rtimes\Gamma\) for some profinite actions, and S. Popa and S. Vaes [Acta Math. 212, No. 1, 141–198 (2014; Zbl 1307.46047)] succeeded in removing the profiniteness assumption which allows to give examples of weakly amenable groups \(\Gamma\) which are \(\mathcal{C}\)-rigid, i.e., such that any free and ergodic action gives rise to a \(\text{II}_1\) factor with a unique Cartan subalgebra, up to unitary conjugacy.

One of the major achievement of the paper under review is to get rid of the weak amenability assumption on \(\Gamma\) and still provide uniqueness of Cartan subalgebra results. It is shown that any group \(\Gamma=\Gamma_1*_\Lambda\Gamma_2\) with \([\Gamma_1:\Lambda]\geq 2\), \([\Gamma_2:\Lambda]\geq 3\) and \(\Lambda\) weakly malnormal in \(\Gamma\) is \(\mathcal{C}\)-rigid (Theorem 1.1). In particular, \(\text{SL}_3(\mathbb{Z})*\mathbb{Z}\) is \(\mathcal{C}\)-rigid and non-weakly amenable. Moreover, using S. Popa’s cocycle superrigidity theorem the author concludes that the Bernoulli action of a direct product of two groups as in Theorem 1.1 is \(W^*\)-superrigid (Corollary 1.2).

To show such a result, the author writes the group measure space construction \(\text{L}^\infty(X)\rtimes\Gamma\) with \(\Gamma=\Gamma_1*_\Lambda\Gamma_2\) as a von Neumann algebraic amalgamated free product and studies, in general, the normalizer of an arbitrary diffuse von Neumann subalgebra \(A\) in an amalgamated free product \(M=M_1*_B M_2\) of tracial von Neumann algebras. The more general type of results which can be found in the paper is the following: if \(A\) is amenable relative to \(B\), \(A\nprec_M B\) and \(P'\cap M^\omega=\mathbb C1\), for some free ultrafilter \(\omega\) (where \(P\) is the von Neumann algebra generated by the normalizer of \(A\) in \(M\)), then either \(P\) is amenable relative to \(B\) or \(P\prec_M M_i\) for some \(i\in\{1,2\}\) (Theorem 1.6). The condition \(P'\cap M^\omega\) is not optimal and has been removed in a subsequent work by S. Vaes [Publ. Res. Inst. Math. Sci. 50, No. 4, 695–721 (2014; Zbl 1315.46067)].

Theorem 1.6 has applications in the study of Cartan subalgebras in a general amalgamated free product of tracial von Neumann algebras (Theorem 1.3). In particular, in the context of equivalence relations, it is shown that, for \(\mathcal{R}_1\) and \(\mathcal{R}_2\), two countable ergodic probability measure preserving equivalence relations on a standard probability space \((X,\mu)\) with infinite orbits almost everywhere and without any hyperfinite restrictions, if \(\mathcal{R}=\mathcal{R}_1*\mathcal{R}_2\) is strongly ergodic, then the von Neumann algebra of \(\mathcal{R}\) has a unique Cartan subalgebra (Corollary 1.4). In the context of free products without amalgamation, it is shown that \(M=M_1*M_2\) does not have any Cartan subalgebra whenever \(M_1\) and \(M_2\) are non-trivial tracial von Neumann algebras with \(\text{dim}(M_1)+\text{dim}(M_2)\geq 5\) (Corollary 1.5). In the context of strong solidity, Theorem 1.6 allows to show that the class of strongly solid von Neumann algebras is closed under free products and a large class of amalgamated free products (Theorem 1.8).

The paper contains more general statements and some other results. It also contains an appendix written by A. Ioana and S. Vaes in which S. Popa’s \(w\)-spectral gap property is discussed. It is shown that the technical assumption used in the proof of Theorem 1.6 is actually equivalent to \(w\)-spectral gap.

Reviewer: Pierre Fima (Paris)

##### MSC:

46L10 | General theory of von Neumann algebras |

46L36 | Classification of factors |

37A20 | Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations |

28D15 | General groups of measure-preserving transformations |