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

Baruch Solel, Mansi Suryawanshi

TL;DR

The paper advances the structural theory of multivariable isometric covariant representations by establishing a general Wold decomposition criterion for tuples $(\sigma,T_1,\dots,T_n)$ of $C^*$-correspondences, introduces and analyzes twisted and doubly twisted product-system representations, and provides explicit Fock-space models for the resulting summands. A complete Wold decomposition is proved in the doubly twisted setting, recovering classical results in special cases, and wandering-subspace technology yields concrete descriptions of the summands. A key contribution is a direct-limit construction that yields unitary extensions for broad families of representations, with notable applications to doubly twisted, doubly non-commuting, and doubly commuting tuples, including multiplication on $H^2(\mathbb D^n)$. Together, these results supply a unified framework for canonical decompositions and functional models in high-rank operator theory.

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 $(σ, T_1, T_2, \ldots, T_n)$, where each $(σ,T_i)$ 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.

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

TL;DR

The paper advances the structural theory of multivariable isometric covariant representations by establishing a general Wold decomposition criterion for tuples of -correspondences, introduces and analyzes twisted and doubly twisted product-system representations, and provides explicit Fock-space models for the resulting summands. A complete Wold decomposition is proved in the doubly twisted setting, recovering classical results in special cases, and wandering-subspace technology yields concrete descriptions of the summands. A key contribution is a direct-limit construction that yields unitary extensions for broad families of representations, with notable applications to doubly twisted, doubly non-commuting, and doubly commuting tuples, including multiplication on . Together, these results supply a unified framework for canonical decompositions and functional models in high-rank operator theory.

Abstract

We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of -correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple , where each is an isometric covariant representation of a \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.
Paper Structure (8 sections, 47 theorems, 266 equations)

This paper contains 8 sections, 47 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterization of the Wold Decomposition
  4. Doubly Twisted Representations
  5. Examples
  6. The Wold Decomposition
  7. Model Operators on the Fock Space
  8. Unitary Extensions

Key Result

Theorem 2.5

MS Let $(\sigma,T)$ be an isometric covariant representation of a $C^*$-correspondence $E$ over a $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$. Then there exists a unique orthogonal decomposition such that The decomposition is unique in the sense that if a closed subspace $\mathcal{K}\subseteq\mathcal{H}$ reduces $(\sigma,T)$ and $\left.(\sigma,T)\right|_{\mathcal{K}}$ is induced

Theorems & Definitions (108)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 98 more