Extensions and Dilations for $C^*$-dynamical Systems

Abstract

Let $A$ be a unital $C^*$-algebra and $α$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,α)$ is a pair $(π,T)$ consisting of a $C^*$-representation $π$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $Tπ(α(a))=π(a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(ρ,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(σ,U)$ (on a larger space $K_1$) with $U$ unitary. Our objective here is to give self-contained, elementary proofs of these results which avoid the technology of $C^*$-correspondences. We also discuss the non uniqueness of the extension.