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

Abstract

Let $(G,\Lambda )$ be a self-similar $k$-graph with a possibly infinite vertex set ${\Lambda }^0$. We associate a universal C*-algebra ${\mathcal{O}}_{G,\Lambda }$ to $(G,\Lambda )$. The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\Lambda }$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of ${\Lambda }^0$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\Lambda }$. Under some conditions, we characterize all primitive ideas of ${\mathcal{O}}_{G,\Lambda }$. 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.

Cited by in corpus (1)

• Remarks On the K-theory of C*-Algebras of Products of Odometers