A question about a conditional expectation in C*-algebra Let $\Gamma$ be a discrete group. Consider a conditional expectation $\Phi: B(l^{2}(\Gamma))\rightarrow l^{\infty}(\Gamma)$ defined by
$$\Phi(T)=\sum_{g\in \Gamma}e_{g,g}Te_{g,g},$$
where $e_{g,g}$ is the rank-one projection onto $\delta_{g}\in l^{2}(\Gamma)$ (where $\{\delta_{t} : t\in \Gamma\}$ is the canonical orthonormal basis) and the sum is taken in the strong operator topology.
Then, can we verify that $\Phi(\lambda_{s}T\lambda_{s}^{*})=\lambda_{s}\Phi(T)\lambda_{s}^{*}$ ? (Here, the $\lambda_{s}$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, t\in \Gamma$.) 

Note that $\Phi$ has norm one and $\Phi\circ \Phi = \Phi$. Then use Tomiyama's Theorem (Theorem II.6.10.2 from Blackadar's book Operator algebras; thanks Martin!). Actually, the proof of this theorem will show you how to prove the module property.
