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

TL;DR

This paper characterizes the gauge-invariant ideal structure of C*-algebras from proper product systems over ^d, unifying previous parametrizations and applying results to dynamical systems and higher-rank graphs.

Contribution

It establishes a unified gauge-invariant ideal parametrization for C*-algebras associated with proper product systems over ^d, simplifying previous results and extending applications.

Findings

Unified the gauge-invariant ideal parametrization with Bilich's results.

Simplified the main parametrization result for proper product systems.

Applied the findings to C*-dynamical systems and higher-rank graphs.

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.