representation* is a representation $W \in M(C^*_\lambda(G) \otimes C(G))$ of ${\hat{G}}$ on $C(G)$ given by $W(g) :=\lambda_g^*$ regarding $M(C^*_\lambda(G) \otimes C(G))$ as the ${\mathrm{C}^*}$-algebra of $M(C^*_\lambda(G))$-valued strictly continuous functions on $G$.

There is a natural one-to-one correspondence between representations of ${\hat{G}}$ on $A$ and unital $\ast$-homomorphisms $C(G) \to M(A)$ as following:

-   For a representation $U$ of ${\hat{G}}$ on $A$, the representation $\pi_U(f) := (f \otimes {\mathrm{id}})(U) \in M(A)$ of $B(G) = C^*_\lambda(G)^* \subset C(G)$ extends to a $\ast$-homomorphism of $C(G)$.

-   For a unital $\ast$-homomorphism $\pi$ of $C(G)$ on $M(A)$, then $U_\pi := ({\mathrm{id}}\otimes \pi)(W)$ is a representation of ${\hat{G}}$.

A (left) *action* of ${\hat{G}}$ on a ${\mathrm{C}^*}$-algebra $A$ is a nondegenerate $\ast$-homomorphism $\alpha : A \to  M(C^*_\lambda(G) \otimes A)$ such that $$({\mathrm{id}}\otimes \alpha)\alpha = ({\hat \Delta}\otimes {\mathrm{id}}) \alpha .$$ For example, for any $G$-${\mathrm{C}^*}$-algebra $A$, the crossed product $G \ltimes A$ is a ${\hat{G}}$-${\mathrm{C}^*}$-algebra by the $*$-homomorphism ${\hat \alpha}: G \ltimes A \to M(C^*_\lambda(G) \otimes (G \ltimes A))$ uniquely determined by ${\hat \alpha}(a) = a$ and ${\hat \alpha}(u_g) = u_g \otimes u_g$ for $a \in A$, $g \in G$.

For an action $\alpha$ of ${\hat{G}}$ on $A$, the reduced *crossed product* is defined by $${\hat{G}}{\, }_{\alpha}\! \ltimes A 