On unitary groups of crossed product von Neumann algebras

Abstract

We consider the tracial crossed product algebra M=A⋊Λ arising from a trace preserving action σ:Λ↷A of a discrete group Λ on a tracial von Neumann algebra A. For a unitary subgroup G⊂U(M), we study when this G can be conjugated into U(A)⋅Λ in M. We provide a general sufficient condition for this to happen. Our result generalizes [14, Theorem 3.1] which treats the case when M is the group von Neumann algebra L(Λ).