Covariant representations of subproduct systems

TL;DR

This paper generalizes Pimsner's theorem to subproduct systems, providing conditions under which covariant representations extend to $C^*$-representations, and explores related dilation and decomposition results.

Contribution

It establishes sufficient and necessary conditions for covariant representations of subproduct systems to extend to $C^*$-representations, broadening the scope beyond classical cases.

Findings

Identifies conditions for extension to $C^*$-representations

Proves universality of the tensor algebra

Develops dilation and Wold decomposition results

Abstract

A celebrated theorem of Pimsner states that a covariant representation $T$ of a $C^*$-correspondence $E$ extends to a $C^*$-representation of the Toeplitz algebra of $E$ if and only if $T$ is isometric. This paper is mainly concerned with finding conditions for a covariant representation of a \emph{subproduct system} to extend to a $C^*$-representation of the Toeplitz algebra. This framework is much more general than the former. We are able to find sufficient conditions, and show that in important special cases, they are also necessary. Further results include the universality of the tensor algebra, dilations of completely contractive covariant representations, Wold decompositions and von Neumann inequalities.