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The ideal structures of self-similar $k$-graph C*-algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  11 June 2020

HUI LI  and
DILIAN YANG
HUI LI
Affiliation:
Department of Mathematics and Physics, North China Electric Power University, Beijing102206, China (e-mail: lihui8605@hotmail.com)
DILIAN YANG
Affiliation:
Department of Mathematics & Statistics, University of Windsor, Windsor, OntarioN9B 3P4, Canada (e-mail: dyang@uwindsor.ca)
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Abstract

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$-graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$. We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$. The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$-graph C*-algebras in depth.

Keywords

self-similar k-graphC*-algebragauge-invariant idealprimitive idealUniversal Coefficient Theorem

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2480 - 2507
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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