Non-cyclotomic Presentations of Modules and Prime-order Automorphisms of Kirchberg Algebras

TL;DR

This paper demonstrates that prime-order automorphisms of K-theory for UCT Kirchberg algebras can be lifted to automorphisms of the algebras themselves, using novel module presentations without cyclotomic summands.

Contribution

It introduces a new method for representing modules over cyclic group rings, enabling the lifting of automorphisms from K-theory to algebra automorphisms in Kirchberg algebras.

Findings

Automorphisms of prime order in K-theory are induced by algebra automorphisms.

Modules over integral group rings of cyclic groups have presentations with no cyclotomic summands.

Extension of results to equivariant inclusions of Kirchberg algebras.

Abstract

We prove the following theorem: let $A$ be a UCT Kirchberg algebra, and let $\alpha$ be a prime-order automorphism of $K_*(A)$, with $\alpha([1_A])=[1_A]$ in case $A$ is unital. Then $\alpha$ is induced from an automorphism of $A$ having the same order as $\alpha$. This result is extended to certain instances of an equivariant inclusion of Kirchberg algebras. As a crucial ingredient we prove the following result in representation theory: every module over the integral group ring of a cyclic group of prime order has a natural presentation by generalized lattices with no cyclotomic summands.