Ideal structure of Nica-Toeplitz algebras

TL;DR

This paper characterizes the gauge-invariant ideal structure of Nica-Toeplitz algebras associated with product systems over b^n, providing a classification in terms of ideals in the underlying algebra and applying it to higher-rank graphs.

Contribution

It offers a comprehensive description and classification of gauge-invariant ideals in Nica-Toeplitz algebras for proper product systems, extending understanding in higher-rank graph contexts.

Findings

Classified gauge-invariant ideals in b^n product systems.

Described restrictions of gauge-invariant ideals to the base algebra.

Applied the classification to higher-rank graph algebras.

Abstract

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra $\mathcal{NT}(X)$ of a product system $(A, X)$ over $\mathbb{N}^n$. We obtain a clear description of $X$-invariant ideals in $A$, that is, restrictions of gauge-invariant ideals in $\mathcal{NT}(X)$ to $A$. The main result is a classification of gauge-invariant ideals in $\mathcal{NT}(X)$ for a proper product system in terms of families of ideals in $A$. We also apply our results to higher-rank graphs.