1 results
On invariant subalgebras of group
$C^*$ and von Neumann algebras
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 04 November 2022, pp. 3341-3353
- Print publication:
- October 2023
-
- Article
- Export citation
-
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.
