The inner structure of boundary quotients of right LCM semigroups

TL;DR

This paper investigates the structure and automorphisms of boundary quotients from algebraic dynamical systems, providing a complete extension of Bogolubov automorphisms and characterizing automorphisms fixing certain subalgebras.

Contribution

It offers a comprehensive analysis of automorphisms of boundary quotients, including extending Bogolubov automorphisms and describing automorphisms fixing specific subalgebras.

Findings

Automorphisms fixing $C^*(G)$ form a maximal abelian subgroup.

Every automorphism fixing a natural Cuntz subalgebra is a gauge automorphism.

Many automorphisms are shown to be outer.

Abstract

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.