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

TL;DR

This paper introduces convex subshifts as a framework for partial actions of free groups on totally disconnected spaces, and analyzes the ideal structure of associated tame separated graph algebras, linking algebraic properties to combinatorial graph data.

Contribution

It defines convex subshifts, relates finite type subshifts to finite bipartite separated graphs, and characterizes the ideal structure of tame separated graph algebras via separated Bratteli diagrams.

Findings

Lattice of ideals corresponds to hereditary saturated subsets of the separated Bratteli diagram.

Characterization of simplicity and primeness of tame separated graph algebras.

Establishment of a framework connecting partial group actions, graph algebras, and combinatorial structures.

Abstract

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, $\mathcal{O}(E,C)$ and $\mathcal{O}^r(E,C)$, of a separated graph $(E,C)$, and of the abelianized Leavitt path algebra $L_K^{\text{ab}}(E,C)$ 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 $D^{\infty}$-saturated subsets of a certain infinite separated graph $(F_{\infty},D^{\infty})$ 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.