Ideal Structure of Nica-Toeplitz Algebras

Abstract

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra \(\mathcal {N}\mathcal {T}(X)\) 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 \(\mathcal {N\hspace{-1.111pt}T}(X)\) to A. The main result is a classification of gauge-invariant ideals in \(\mathcal {N\hspace{-1.111pt}T}(X)\) for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.