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Gauge-invariant ideal structure of C*-algebras associated with proper product systems over $\mathbb{Z}_+^d$

Joseph A. Dessi

Abstract

We show that the gauge-invariant ideal parametrisation results of the author and Kakariadis are in agreement with those of Bilich in the case of a proper product system over $\mathbb{Z}_+^d$. This is accomplished in two ways: first via the use of Nica-covariant representations and Gauge-Invariant Uniqueness Theorems (the indirect route), and second via the definitions of the parametrising objects alone (the direct route). We then apply our findings to simplify the main parametrisation result of the author and Kakariadis in the proper case, thereby fully describing the gauge-invariant ideal structure of each equivariant quotient of the Toeplitz-Nica-Pimsner algebra. We close by providing applications in the contexts of C*-dynamical systems and row-finite higher-rank graphs.

Gauge-invariant ideal structure of C*-algebras associated with proper product systems over $\mathbb{Z}_+^d$

Abstract

We show that the gauge-invariant ideal parametrisation results of the author and Kakariadis are in agreement with those of Bilich in the case of a proper product system over . This is accomplished in two ways: first via the use of Nica-covariant representations and Gauge-Invariant Uniqueness Theorems (the indirect route), and second via the definitions of the parametrising objects alone (the direct route). We then apply our findings to simplify the main parametrisation result of the author and Kakariadis in the proper case, thereby fully describing the gauge-invariant ideal structure of each equivariant quotient of the Toeplitz-Nica-Pimsner algebra. We close by providing applications in the contexts of C*-dynamical systems and row-finite higher-rank graphs.

Paper Structure

This paper contains 19 sections, 37 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Description of main results
  5. Contents of sections
  6. C*-correspondences and product systems
  7. Notation
  8. C*-correspondences
  9. Product systems
  10. Gauge-invariant ideal structure of ${\mathcal{N}}{\mathcal{T}}_X$
  11. NT-$2^d$-tuples
  12. T-families
  13. Connection between NT-$2^d$-tuples and T-families
  14. NT-$2^d$-tuples are T-families
  15. T-families are NT-$2^d$-tuples
  16. ...and 4 more sections

Key Result

Lemma 2.2.1

Lan95 Let $X$ and $Y$ be C*-correspondences over a C*-algebra $A$. For $x\in X$, the equation $\Theta_x(y)=x\otimes y \; (y\in Y)$ defines an element $\Theta_x\in{\mathcal{L}}(Y,X\otimes_AY)$ satisfying $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (68)

  • Lemma 2.2.1
  • Theorem 2.2.2
  • Lemma 2.2.3
  • Theorem 2.2.4
  • Proposition 2.3.1
  • proof
  • Proposition 2.3.2
  • Proposition 2.3.3
  • Proposition 2.3.4
  • Proposition 2.3.5
  • ...and 58 more