On unitary groups of crossed product von Neumann algebras

We consider the tracial crossed product algebra $M=A⋊\mathrm{\Lambda }$ arising from a trace preserving action $\sigma :\mathrm{\Lambda }\curvearrowright A$ of a discrete group $\mathrm{\Lambda }$ on a tracial von Neumann algebra $A$. For a unitary subgroup $\mathcal{G}\subset \mathcal{U}(M)$, we study when this $\mathcal{G}$ can be conjugated into $\mathcal{U}(A)\cdot \mathrm{\Lambda }$ in $M$. We provide a general sufficient condition for this to happen. Our result generalizes a result of Ioana, Popa and Vaes, which treats the case when $M$ is the group von Neumann algebra $L(\mathrm{\Lambda })$.