C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

Abstract

A X-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In Doc. Math. 7 (2002) 1-30, the author constructed a C-algebra 0 £ associated with a X-graph system £ from a graph theoretic view-point. If a X-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebra 0 £ , under a certain condition on £ called (II). As a result, the class of the C*-algebras associated with X-graph systems under condition (II) is closed under quotients by its ideals.2000 Mathematics subject classification: primary 46L35; secondary 46L05, 37B10.

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

“…The ideal structure of C * -algebras of Hilbert C * -bimodules has been studied in Kajiwara et al [13] (cf. [18,27]). Their paper is written in the language of Hilbert C * -bimodules.…”

Subshifts and 𝐶*-algebras from one-counter codes

“…Proof. There exists a bijective correspondence between hereditary subsets of the vertex set V and ideals in the C * -algebra O L ( [18], [20]).…”

𝐶*-algebras, groupoids and covers of shift spaces

“…In [55,Section 4.3] (see also [16,Corollary 6.7]), Thomsen realized it as a groupoid C * -algebra of a semi-étale groupoid, Carlsen and Silvestrov describe it as one of Exel's crossed products [15,Theorem 10], while Dokuchaev and Exel use partial actions [21,Theorem 9.5]. Matsumoto then took a slightly different approach and considered certain labeled Bratteli diagrams called λ-graph systems and associated to each λ-graph system L a C * -algebra O L [33][34][35]. Any two-sided subshift Λ has a canonical λ-graph system L Λ and the spectrum of the diagonal subalgebra of O L is (homeomorphic to) the λ-graph L Λ .…”