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

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.

Exel-Pardo algebras of self-similar $k$-graphs

“…In this section, we associate a * -algebra to a self-similar k-graph as the algebraic analogue of [15,Definition 3.9]. Let us first review some definitions and notations.…”

“…Note that although only finite graphs are considered in [6], but many of arguments and results may be easily generalized for countable row-finite graphs with no sources (see [7,9] for example). Inspired from [6], Li and Yang in [14,15] introduced self-similar action of a discrete countable group G on a rowfinite k-graph Λ. They then associated a universal C * -algebra O G,Λ to (G, Λ) satisfying specific relations.…”

“…In particular, we investigate the ideal structure, where our results are new even in the 1-graph case. Our algebras are the algebraic analogue of O G,Λ [14,15], and include many important known algebras such as the Exel-Pardo * -algebras L R (G, E) [5,8], algebraic Katsura algebras [8], Kumjian-Pask algebras [1], and the quotient boundary algebras Q alg R (Λ ⊲⊳ G) of a Zappa-Szép product Λ ⊲⊳ G introduced in Section 3.…”

The Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras

“…In this subsection, we recall the background of k-graph C*-algebras and self-similar k-graph C*-algebras from [1,7,8,9]. Definition 1.1.…”

“…Define ϕ:= π • q • ρ • ι −1 . Since (G,E) is aperiodic, by [10, Proposition 2.66] and by[9, Theorem 3.19] the induced representation Ind − ϕ is injective on O G,E . So ϕ is injective.…”