Unitary representation of amenable groups

Question

Suppose $\Gamma$ is an amenable group, let $\pi:\Gamma\mapsto \mathcal{u}(\mathcal{H})$ and $\rho:\Gamma\mapsto \mathcal{u}(\mathcal{K})$ are two unitary representations of $\Gamma$, what does one mean by $T \in B(\mathcal{H,K})$ is $\Gamma$ invariant? I don't get the meaninng.

Answer 1

Well it should be $\Gamma$ invariant w.r.t. $\pi$ (or w.r.t. $\rho$). $T$ being $\Gamma$ invariant w.r.t. $\pi$ means that $T(v) = T(\pi(g)v)$ for each $g \in G$.