Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions

TL;DR

This paper develops a comprehensive operator-theoretic framework for Wold decompositions and unitary extensions of multivariable isometric covariant representations of product systems of $C^*$-correspondences, including twisted variants.

Contribution

It introduces twisted and doubly twisted covariant representations, proves Wold decomposition existence for these, and provides explicit models and extension procedures.

Findings

Wold decomposition exists for doubly twisted isometric representations.

Explicit Fock-type models for Wold summands are constructed.

Unitary extensions are achieved via a direct-limit approach.

Abstract

We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of $C^*$-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple $(\sigma, T_1, T_2, \ldots, T_n)$, where each $(\sigma,T_i)$ is an isometric covariant representation of a $C^*$\nobreakdash-correspondence. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of product system.