Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies

TL;DR

This paper investigates the conditions under which inclusions of certain algebraic structures induce homomorphisms of their associated C*-algebras and demonstrates that K-theory remains invariant under homotopic cocycles in specific self-similar groupoid actions.

Contribution

It establishes criteria for induced homomorphisms of twisted C*-algebras and proves K-theory invariance for a class of self-similar groupoid actions on k-graphs.

Findings

Inclusions of finitely aligned categories can induce twisted C*-algebra homomorphisms under certain conditions.

An example shows some inclusions do not induce homomorphisms between Toeplitz algebras.

K-theory is invariant for twisted C*-algebras of self-similar groupoid actions with homotopic cocycles.

Abstract

We establish conditions under which an inclusion of finitely aligned left-cancellative small categories induces inclusions of twisted C*-algebras. We also present an example of an inclusion of finitely aligned left-cancellative monoids that does not induce a homomorphism even between (untwisted) Toeplitz algebras. We prove that the twisted C*-algebras of a jointly faithful self-similar action of a countable discrete amenable groupoid on a row-finite k-graph with no sources, with respect to homotopic cocycles, have isomorphic K-theory.