Covariant representations of subproduct systems: Invariant subspaces and curvature

TL;DR

This paper characterizes invariant subspaces of covariant representations of subproduct systems, extends the notion of curvature for these representations, and derives consequences like a Beurling type theorem and properties of wandering subspaces.

Contribution

It provides a new characterization of invariant subspaces in covariant representations and extends curvature concepts, connecting them with invariant subspace structure.

Findings

Invariant subspaces correspond to partial isometries with specific properties.

Extension of curvature notions to covariant representations.

Derivation of a Beurling type theorem and analysis of wandering subspaces.

Abstract

Let $X=(X(n))_{n \in \mathbb{Z_+}}$ be a standard subproduct system of $C^*$-correspondences over a $C^*$-algebra $\mathcal M.$ Assume $T=(T_n)_{n \in \mathbb{Z_+}}$ to be a pure completely contractive, covariant representation of $X$ on a Hilbert space $\mathcal H,$ and $\mathcal S$ to be a non-trivial closed subspace of $\mathcal H.$ Then $\mathcal{S}$ is invariant for $T$ if and only if there exist a Hilbert space $\mathcal{D},$ a representation $\pi$ of $\mathcal M$ on $\mathcal D,$ and a partial isometry $\Pi: \mathcal{F}_X\bigotimes_{\pi}\mathcal{D}\to \mathcal{H} $ such that $$\Pi (S_n(\zeta)\otimes I_{\mathcal{D}})=T_n(\zeta)\Pi~\mbox{whenever}~\zeta\in X(n), ~n\in \mathbb{Z_+},~\mbox{and}$$ $\mathcal S$ is the range of $\Pi,$ or equivalently, $P_{\mathcal S}=\Pi\Pi^*.$ This result leads us to many important consequences including Beurling type theorem and other general observations on wandering subspaces. We extend the notion of curvature for completely contractive, covariant representations and analyze it in terms of the above results.