Abstract. We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action Γ G, whenever Γ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd [BG06, BG10], and Benoist and de Saxcé [BdS14]. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique Γ-invariant finitely additive measure defined on all bounded measurable subsets of G.
Abstract. We prove that there exist uncountably many separable II1 factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II1 factors originally introduced by McDuff [MD69a,MD69b] are such examples. This entails the existence of a continuum of non-elementarily equivalent II1 factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
Abstract. We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms of Λ-invariant measures on some compact Γ-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C * -algebra.
We show that stationary characters on irreducible lattices $\Gamma < G$
Γ
<
G
of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$
Γ
<
G
, the left regular representation $\lambda _{\Gamma }$
λ
Γ
is weakly contained in any weakly mixing representation $\pi $
π
. We prove that for any such irreducible lattice $\Gamma < G$
Γ
<
G
, any Uniformly Recurrent Subgroup (URS) of $\Gamma $
Γ
is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$
Γ
<
G
. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.
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