We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$C^*$ and von Neumann algebras
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.
