Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Abstract

In this paper, we investigate several structural properties for crossed product II1 factors M arising from free Bogoljubov actions associated with orthogonal representations π : G → O(HR) of arbitrary countable discrete groups. Under fairly general assumptions on the orthogonal representation π : G → O(HR), we show that M does not have property Gamma of Murray and von Neumann. Then we show that any regular amenable subalgebra A ⊂ M can be embedded into L(G) inside M . Finally, when G is assumed to be amenable, we locate precisely any possible amenable or Gamma extension of L(G) inside M .2010 Mathematics Subject Classification. 46L10; 46L54; 46L55; 22D25.

Thin II1 factors with no Cartan subalgebras

“…The main result of this section is the following dichotomy theorem for A-valued semicircular systems. In the special case of free Bogoljubov crossed products (see Remark 3.5), this result was proven in [Ho12b,Theorem B]. As explained in the introduction, the A-valued semicircular systems fit perfectly into Popa's deformation/rigidity theory.…”

“…In Theorem 5.1, we prove several maximal amenability results for the inclusion A ⊂ M associated with a symmetric A-bimodule (H, J), by combining the methods of [Po83,BH16]. Again, these results generalize [Ho12a,Ho12b] where the same was proved for free Bogoljubov crossed products.…”

“…The point of view of A-valued semicircular systems is more flexible and even offers advantages in the study of free Bogoljubov crossed products M = L(F ∞ ) ⋊ G. Indeed, in Corollary 6.4, we prove that these II 1 factors M never have a Cartan subalgebra, while in [Ho12b], this could only be proved for special classes of orthogonal representations.…”

The cup subalgebra has the absorbing amenability property

“…Popa introduced the notion of asymptotic orthogonality property (AOP) in [Pop83]. We consider a strengthening of this notion which was used by Houdayer and the second author [Hou14b,Wen]. Definition 1.1.…”

“…Inspired by this question and the work of Houdayer on maximal Gamma extensions [Hou14b,Hou15], we consider the notion of absorbing amenability property (AAP). An inclusion of von Neumann algebras A ⊂ M has the AAP if for any diffuse subalgebra B ⊂ A and any amenable intermediate algebra B ⊂ D ⊂ M we have that D is contained in A.…”