Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

Pere Ara, Matias Lolk

Research output: Contribution to journalArticleResearchpeer-review

Abstract

© 2018 Elsevier Inc. We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algebras, O(E,C) and Or(E,C), of a separated graph (E,C), and of the abelianized Leavitt path algebra LKab(E,C) as well. These algebras are the (reduced) crossed products with respect to the above-mentioned partial actions, and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary D∞-saturated subsets of a certain infinite separated graph (F∞,D∞) built from (E,C), called the separated Bratteli diagram of (E,C). We finally use these tools to study simplicity and primeness of the tame separated graph algebras.
Original languageEnglish
Pages (from-to)367-435
JournalAdvances in Mathematics
Volume328
DOIs
Publication statusPublished - 13 Apr 2018

Keywords

  • Ideal structure
  • Primeness
  • Separated graph
  • Simplicity
  • Tame graph algebra

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