Rigidity for Von Neumann Algebras Given by Locally Compact Groups and Their Crossed Products

Abstract

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W * strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.

The approximation property and exactness of locally compact groups

“…Recent researches (see e.g., [1], [2], [21], [27], [32], [33]) show that non-discrete locally compact groups also provide rich sources of interesting operator algebras. They also reveal attractive and fruitful interactions between locally compact group theory and theory of operator algebras.…”

Ozawa's class𝒮for locally compact groups and unique prime factorization of group von Neumann algebras

“…C, but we first need the following equivalent characterizations of the existence of a map satisfying (1.1). Note that point (i) in the proposition below is property (S) in the sense of [6].…”

“…Proof of ?? E. As mentioned in the Introduction exactness is a measure equivalence invariant by [14, corollary 2.9] and [13, theorem 0.1 (6)]. The characterization of measure equivalence in terms of stable isomorphism of cross-section equivalence relations (see [27, theorem A] and [26, theorem A]) together with ??…”

“…The following result is a locally compact version of [33, A notion of measure equivalence for locally compact groups was introduced by S. Deprez and Li in [13]. By [14, corollary 2.9] and [13, theorem 0.1 (6)] exactness is preserved under this notion of measure equivalence. More recently, this notion has been studied in more detail in [26,27].…”

“…In [6], the author proved together with Brothier and Vaes that the group von Neumann algebra L(G) is solid whenever G is a locally compact group in class S. In particular, if L(G) is also a non-amenable factor, then L(G) is prime. Combining ??…”