Ideal Structure of Nica-Toeplitz Algebras

Abstract

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}\\mathcal {T}(X)$$\\end{document} of a product system (A, X) over Nn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}^n$$\\end{document}. We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} to A. The main result is a classification of gauge-invariant ideals in NT(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N\\hspace{-1.111pt}T}(X)$$\\end{document} for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.