On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

TL;DR

This paper proves that certain automorphisms of simple Cuntz-Krieger algebras, specifically diagonal quasi-free automorphisms with an aperiodic matrix, are strongly approximately inner, expanding understanding of their symmetry properties.

Contribution

It establishes that outer actions by finite abelian groups via diagonal quasi-free automorphisms are strongly approximately inner under aperiodicity conditions, using approximate cohomology techniques.

Findings

Outer actions are strongly approximately inner under specified conditions

The result applies to automorphisms given by diagonal quasi-free automorphisms

The proof involves an approximate cohomology vanishing argument

Abstract

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.