Divisibility properties for C*-algebras

TL;DR

This paper explores divisibility properties in the Cuntz semigroup of C*-algebras, develops construction methods for algebras with specific divisibility traits, and demonstrates the existence of a sequence of simple C*-algebras whose product has a character.

Contribution

It introduces three notions of divisibility in the Cuntz semigroup and provides methods to construct C*-algebras with prescribed divisibility behaviors.

Findings

Existence of a sequence of simple unital infinite-dimensional C*-algebras with a product having a character

Development of methods to construct C*-algebras with specific divisibility properties

Analysis of how divisibility notions reflect properties of C*-algebras

Abstract

We consider three notions of divisibility in the Cuntz semigroup of a C*-algebra, and show how they reflect properties of the C*-algebra. We develop methods to construct (simple and non-simple) C*-algebras with specific divisibility behaviour. As a byproduct of our investigations, we show that there exists a sequence $(A_n)$ of simple unital infinite dimensional C*-algebras such that the product $\prod_{n=1}^\infty A_n$ has a character.