Extensions and Dilations for $C^{\ast }$-dynamical Systems

Abstract

Let $A$ be a unital $C^{\ast }$-algebra and $\alpha$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,\alpha )$ is a pair $(\pi ,T)$ consisting of a $C^{\ast }$-representation $\pi$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $T\pi (\alpha (a))=\pi (a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(\rho ,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(\sigma ,U)$ (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 $C^{\ast }$-correspondences. We also discuss the non uniqueness of the extension.

11 pages

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