A Kurosh-type theorem for type II\(_1\) factors.

*(English)*Zbl 1114.46041One may assume that the von Neumann algebras referred to in this article are ultraweak closures of exact C*-algebras, as for instance are the group von Neumann algebras of discrete exact groups. All tensor products referred to are von Neumann tensor products. The author’s §3 deals with Kurosh’s theorem and tensor products of the group von Neumann algebras. His §4 is concerned with crossed products and solidity results. A von Neumann algebra is called diffuse if it has no minimal projections. A type II\(_{1}\) factor is prime if it is not isomorphic to a tensor product of any two type II\(_{1}\) factors. e.g., group factors for free groups are prime. A type II\(_{1}\) factor which is injective is also prime.

A. Kuroš [Math. Ann. 109, 647–660 (1934; JFM 60.0082.02)] proved that for a sequence of groups \(\{ \Gamma_{i} \}\) their free product can be expressed as a free product of a free group and conjugates of the \(\Gamma_{i}\). In particular every subgroup \(\Lambda\) of a free product \(\Gamma_{1} * \Gamma_{2}\) of groups is itself a free product of a free group \(F\) and conjugates of subgroups of \(\Gamma_{i}\) or \(\Gamma_{2}\). The author proves an analogous result for the free product \(M\) of von Neumann algebras \(M_{i}\), viz., if \(N\) is a non-prime non-injective subfactor of the free product \(M_{1} * M_{2}\), whose relative commutant is a factor, then there exists a unitary \(u \in M\) such that \(u^{*}Nu\) is contained in \(M_{1}\) or \(M_{2}\). It follows that \(M\) is prime unless one of the \(M_{i}\) is trivial or if both \(M_{i}\) are isomorphic to \(\mathbb{M}_{2}(\mathbb{C})\). The author’s theorem is a bit more general in that it holds also when he takes the above \(N\) to be any injective type II\(_{1}\) subfactor of the free product.

The author calls a von Neumann algebra solid if for any diffuse von Neumann subalgebra the relative commutant is injective, following N. Ozawa [Acta Math. 192, No. 1, 111–117 (2004; Zbl 1072.46040)]. A weak version, for a finite von Neumann algebra, is semi-solidity, viz., the relative commutant of any type II\(_{1}\) subalgebra is injective.

The author considers the class, denoted by \(\mathcal{S}\), of countable discrete groups \(\Gamma\) such that the right and left actions of \(\Gamma \times \Gamma\), on the ‘boundary’ of the Stone-Čech compactification of \(\Gamma\), are amenable. The von Neumann group algebra of such a group \(\Gamma\) is solid. This class of groups is attributed to G. Skandalis [K-Theory 1, No. 6, 549–573 (1988; Zbl 0653.46065)] because of his use of the boundary.

The wreath product of groups \(\Delta\) and \(\Gamma\) involves \(\Gamma\) acting on \(\Delta^{\Gamma}\), a direct sum of \(\Delta\)’s indexed by \(\Gamma\). Here \(\Gamma\) acts on \(\Delta^{\Gamma}\) by left translation, like a Bernoulli-shift action. The author uses the term ‘Bernoulli product’ of group algebras as a von Neumann algebra analogue of the wreath product. He constructs the crossed-product (or covariance) von Neumann algebra for the dynamical system made up of a von Neumann algebra \(A\) with faithful trace \(\tau\) and a trace-preserving action \(\alpha\) as automorphims of a group \(\Gamma \in\mathcal{S}\). The author now follows his methods used to prove solidity in N. Ozawa, loc. cit. He shows that if \(\Gamma\) is exact (which relates to an amenable boundary as in Proposition 4.1) and for copies, indexed by \(\Gamma\), of the hyperfinite type II\(_{1}\) factor \(\mathcal{R}\), for any diffuse von Neumann algebra \(Q \in \bigotimes_{\Gamma}\)\(\mathcal{R}\) the relative commutant of \(Q\) in the Bernoulli product is injective.

He shows also that for a commutative (\(A.\tau)\), \(\Gamma \in\)\(\mathcal{S}\). The crossed product, not necessarily a II\(_{1}\)-factor, is semi-solid. For the commutative von Neumann algebra \(L^{\infty}([0,1])\) and a measure preserving action of \(\Gamma \in {\mathcal S}\), the Bernoulli shift implemented in the crossed product corresponds to the group measure space von Neumann algebra. The author mentions that solid type II\(_{1}\) factors do not necessarily have Murray and von Neumann’s property \((\Gamma)\) but that the latter example, which is semi-solid, does has property \((\Gamma)\), cf., V. Jones’ and K. Schmidt’s [Am. J. Math. 109, 91–114 (1987; Zbl 0638.28014)] result that the cross-product factor does not have the property \((\Gamma)\) if and only if the action is ergodic but not strongly ergodic.

In the case of the von Neumann algebra \(L^{\infty}([0,1])\) and a measure preserving action of \(\Gamma \in \mathcal{S}\) the crossed product corresponds to the group measure space; cf., the Bernoulli or topolopical Markov measure space as constructed in ergodic theory or in Shannon’s information theory.

A. Kuroš [Math. Ann. 109, 647–660 (1934; JFM 60.0082.02)] proved that for a sequence of groups \(\{ \Gamma_{i} \}\) their free product can be expressed as a free product of a free group and conjugates of the \(\Gamma_{i}\). In particular every subgroup \(\Lambda\) of a free product \(\Gamma_{1} * \Gamma_{2}\) of groups is itself a free product of a free group \(F\) and conjugates of subgroups of \(\Gamma_{i}\) or \(\Gamma_{2}\). The author proves an analogous result for the free product \(M\) of von Neumann algebras \(M_{i}\), viz., if \(N\) is a non-prime non-injective subfactor of the free product \(M_{1} * M_{2}\), whose relative commutant is a factor, then there exists a unitary \(u \in M\) such that \(u^{*}Nu\) is contained in \(M_{1}\) or \(M_{2}\). It follows that \(M\) is prime unless one of the \(M_{i}\) is trivial or if both \(M_{i}\) are isomorphic to \(\mathbb{M}_{2}(\mathbb{C})\). The author’s theorem is a bit more general in that it holds also when he takes the above \(N\) to be any injective type II\(_{1}\) subfactor of the free product.

The author calls a von Neumann algebra solid if for any diffuse von Neumann subalgebra the relative commutant is injective, following N. Ozawa [Acta Math. 192, No. 1, 111–117 (2004; Zbl 1072.46040)]. A weak version, for a finite von Neumann algebra, is semi-solidity, viz., the relative commutant of any type II\(_{1}\) subalgebra is injective.

The author considers the class, denoted by \(\mathcal{S}\), of countable discrete groups \(\Gamma\) such that the right and left actions of \(\Gamma \times \Gamma\), on the ‘boundary’ of the Stone-Čech compactification of \(\Gamma\), are amenable. The von Neumann group algebra of such a group \(\Gamma\) is solid. This class of groups is attributed to G. Skandalis [K-Theory 1, No. 6, 549–573 (1988; Zbl 0653.46065)] because of his use of the boundary.

The wreath product of groups \(\Delta\) and \(\Gamma\) involves \(\Gamma\) acting on \(\Delta^{\Gamma}\), a direct sum of \(\Delta\)’s indexed by \(\Gamma\). Here \(\Gamma\) acts on \(\Delta^{\Gamma}\) by left translation, like a Bernoulli-shift action. The author uses the term ‘Bernoulli product’ of group algebras as a von Neumann algebra analogue of the wreath product. He constructs the crossed-product (or covariance) von Neumann algebra for the dynamical system made up of a von Neumann algebra \(A\) with faithful trace \(\tau\) and a trace-preserving action \(\alpha\) as automorphims of a group \(\Gamma \in\mathcal{S}\). The author now follows his methods used to prove solidity in N. Ozawa, loc. cit. He shows that if \(\Gamma\) is exact (which relates to an amenable boundary as in Proposition 4.1) and for copies, indexed by \(\Gamma\), of the hyperfinite type II\(_{1}\) factor \(\mathcal{R}\), for any diffuse von Neumann algebra \(Q \in \bigotimes_{\Gamma}\)\(\mathcal{R}\) the relative commutant of \(Q\) in the Bernoulli product is injective.

He shows also that for a commutative (\(A.\tau)\), \(\Gamma \in\)\(\mathcal{S}\). The crossed product, not necessarily a II\(_{1}\)-factor, is semi-solid. For the commutative von Neumann algebra \(L^{\infty}([0,1])\) and a measure preserving action of \(\Gamma \in {\mathcal S}\), the Bernoulli shift implemented in the crossed product corresponds to the group measure space von Neumann algebra. The author mentions that solid type II\(_{1}\) factors do not necessarily have Murray and von Neumann’s property \((\Gamma)\) but that the latter example, which is semi-solid, does has property \((\Gamma)\), cf., V. Jones’ and K. Schmidt’s [Am. J. Math. 109, 91–114 (1987; Zbl 0638.28014)] result that the cross-product factor does not have the property \((\Gamma)\) if and only if the action is ergodic but not strongly ergodic.

In the case of the von Neumann algebra \(L^{\infty}([0,1])\) and a measure preserving action of \(\Gamma \in \mathcal{S}\) the crossed product corresponds to the group measure space; cf., the Bernoulli or topolopical Markov measure space as constructed in ergodic theory or in Shannon’s information theory.

Reviewer: Aubrey Wulfsohn (Coventry)

##### MSC:

46L10 | General theory of von Neumann algebras |

46L55 | Noncommutative dynamical systems |

46L09 | Free products of \(C^*\)-algebras |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

46L35 | Classifications of \(C^*\)-algebras |