Williams' Conjecture holds for meteor graphs

TL;DR

This paper proves that for meteor graphs, various algebraic and $K$-theoretic invariants are equivalent, confirming Williams' Conjecture in this specific class of graphs.

Contribution

It establishes the equivalence of shift, strong shift, Morita equivalence, and $K$-theory isomorphism for meteor graphs, confirming Williams' Conjecture in this context.

Findings

Meteor graphs' shift and Morita equivalences are equivalent.

Leavitt path algebras of meteor graphs are graded Morita equivalent.

Graph $C^*$-algebras are equivariantly Morita equivalent if their Leavitt path algebras are graded Morita equivalent.

Abstract

A meteor graph is a connected graph with no sources and sinks consisting of two disjoint cycles and the paths connecting these cycles. We prove that two meteor graphs are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K_0^{gr}$, are $\mathbb Z [x,x^{-1}]$-module isomorphic. As a consequence, the Leavitt path algebras of meteor graphs are graded Morita equivalent if and only if their graph $C^*$-algebras are equivariant Morita equivalent.