Ideal structure of Nica-Toeplitz algebras

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.