Definition 41.5.1 (Module Map).label Let $A, B, C$ be $C^{*}$-algebras with $C \subset A, B$ and $1_{A} = 1_{B} = 1_{C}$, then $A$ and $B$ are $C$-bimodules with respect to multiplication.

Let $\phi: A \to B$ be a linear map, then $\phi$ is a $C$-bimodule map if $\phi(c_{1}ac_{2}) = c_{1}\phi(a)c_{2}$ for all $a \in A$ and $c_{1}, c_{2} \in C$.