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Wold-type decomposition for doubly twisted left-invertible covariant representations

Niraj Kumar, Azad Rohilla, Harsh Trivedi

TL;DR

The paper develops a Wold-type decomposition framework for covariant representations of $C^*$-correspondences that are near-isometric and extends it to doubly twisted left-invertible covariant representations on product systems. It introduces a doubly twisted, left-invertible setting with unitary twists $U_{ij}$ and defines key wandering-subspace structures $\mathcal{N}_{\beta}$ to analyze invariant/reducing subspaces. The main result shows that, under suitable per-component decompositions, one can build a multivariate decomposition of the ambient space into $2^m$ reducing subspaces $\mathcal{K}_{\beta}$, such that restricted blocks realize a Wold-type decomposition for subproduct systems $\mathbb{E}_{\beta}$ over $\mathbb{Z}_+^{\beta}$, while the remaining coordinates are induced or invertible. This canonical multivariate structure advances the understanding of covariant representations in product systems and has potential implications for operator-algebraic dynamics and multivariable noncommutative dilation theory.

Abstract

We will introduce the notion of a near-isometric covariant representation of a $C^*$-correspondence and prove its Wold-type decomposition. Wold-type decomposition for doubly twisted left-invertible covariant representations of a product system is also obtained.

Wold-type decomposition for doubly twisted left-invertible covariant representations

TL;DR

The paper develops a Wold-type decomposition framework for covariant representations of -correspondences that are near-isometric and extends it to doubly twisted left-invertible covariant representations on product systems. It introduces a doubly twisted, left-invertible setting with unitary twists and defines key wandering-subspace structures to analyze invariant/reducing subspaces. The main result shows that, under suitable per-component decompositions, one can build a multivariate decomposition of the ambient space into reducing subspaces , such that restricted blocks realize a Wold-type decomposition for subproduct systems over , while the remaining coordinates are induced or invertible. This canonical multivariate structure advances the understanding of covariant representations in product systems and has potential implications for operator-algebraic dynamics and multivariable noncommutative dilation theory.

Abstract

We will introduce the notion of a near-isometric covariant representation of a -correspondence and prove its Wold-type decomposition. Wold-type decomposition for doubly twisted left-invertible covariant representations of a product system is also obtained.
Paper Structure (5 sections, 8 theorems, 59 equations)

This paper contains 5 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Wold-type decomposition for near-isometric covariant representations of C*-correspondences
  4. Basic notations and preparatory results: doubly twisted left-invertible covariant representations
  5. Wold-type decomposition for doubly twisted near-isometry covariant representations of product systems

Key Result

Lemma 1.2

(MR1648483) The map $(\sigma, A)\mapsto \widetilde{A}$ gives a bijection between the set of each completely bounded (resp.completely contractive), covariant representations $(\sigma, A)$ of $E$ on $\mathcal{K}$ and the set of each bounded (resp. contractive) linear maps $\widetilde{A}$ satisfying $\

Theorems & Definitions (27)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • ...and 17 more