On invariant subalgebras of group and von Neumann algebras

Abstract

Given an irreducible lattice $\Gamma $ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma $ -invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal {L}(\Gamma )$ , and for the $\Gamma $ -invariant unital $C^*$ -subalgebras of the reduced group $C^*$ -algebra $C^*_{\mathrm {red}}(\Gamma )$ . We use these results to show that: (i) every $\Gamma $ -invariant von Neumann subalgebra of $\mathcal {L}(\Gamma )$ is generated by a normal subgroup; and (ii) given a weakly mixing unitary representation $\pi $ of $\Gamma $ , every $\Gamma $ -equivariant conditional expectation on $C^*_\pi (\Gamma )$ is the canonical conditional expectation onto the $C^*$ -subalgebra generated by a normal subgroup.

On invariant von Neumann subalgebras rigidity property

“…Let us recall that Γ is said to have Invariant Subgroup Rigidity property (ISR-property) if every Γ-invariant subalgebra M ≤ L(Γ) is of the form L(N) for some normal subgroup N ⊳Γ. This property was studied in the context of higher rank lattices in [AB21,KP22] and in the context of hyperbolic groups in [AJ23,CDS22].…”

Subalgebras, subgroups, and singularity

“…Our strategy is to consider the cases when 𝔼() is trivial or not. In the situation where 𝔼() = ℂ, we want to show that  ⊂ 𝐿(Γ) and use [24,Theorem 1.1] to conclude that  = 𝐿(Λ) for some normal subgroup Λ ⊲ Γ. In the other case, when 𝔼() is nontrivial, we claim that it is enough to show that  ⊂ 𝔼() ⋊ Γ.…”

“…Here, coamenability of  is in the sense of [24], namely, the commutant  ′ ⊂ 𝔹(𝓁 2 (Γ)) admits a Γ-invariant state (similarly for  ⊂ 𝐿(Γ)). Kalantar-Panagopoulos proved the conclusion of Theorem 1.1 for higher rank lattices using "noncommutative Nevo-Zimmer" theorem [5].…”

“…Here, coamenability of $$\mathcal {A}$$ is in the sense of [24], namely, the commutant $$\mathcal {A}^{\prime }\subset \mathbb {B}(\ell ^2(\Gamma ))$$ admits a $$\Gamma$$‐invariant state (similarly for $$\mathcal {M}\subset L(\Gamma )$$). Kalantar–Panagopoulos proved the conclusion of Theorem 1.1 for higher rank lattices using “noncommutative Nevo–Zimmer” theorem [5].…”

“…Now, assume that 𝜏 is not Γ-invariant. Let us observe that the action Γ ↷  is ergodic (cf [24, . Lemma 2.16]…”