Maximal UHF subalgebras of certain C*-algebras

TL;DR

This paper introduces the concept of maximal UHF subalgebras within certain C*-algebras, establishing their existence, uniqueness, and applications to realizing rational subgroups of dimension groups.

Contribution

It defines maximal UHF subalgebras in C*-algebras, proves their existence and uniqueness under specific conditions, and applies this to realize rational subgroups of dimension groups.

Findings

Maximal UHF subalgebras exist and are unique in certain C*-algebras with unperforated K_0-groups.

Not all unital C*-algebras have maximal UHF subalgebras, such as the universal free product M_2 * M_3.

Application to realizing rational subgroups of dimension groups via C*-algebras with maximal UHF subalgebras.

Abstract

A well-known result in dynamical systems asserts that any Cantor minimal system $(X,T)$ has a maximal rational equicontinuous factor $(Y,S)$ which is in fact an odometer, and realizes the rational subgroup of the $K_0$-group of $(X,T)$, that is, $\mathbb{Q}(K_0(X,T), 1) \cong K^0(Y,S)$. We introduce the notion of a maximal UHF subalgebra and use it to obtain the C*-algebraic alonog of this result. We say a UHF subalgebra $B$ of a unital C*-algebra $A$ is a maximal UHF subalgebra if it contains the unit of $A$ any other such C*-subalgebra embeds unitaly into $B$. We prove that if $K_0(A)$ is unperforated and has a certain $K_0$-lifting property, then $B$ exists and is unique up to isomorphism, in particular, all simple separable unital C*-algebras with tracial rank zero and all unital Kirchberg algebras whose $K_0$-groups are unperforated, have a maximal UHF subalgebra. Not every unital C*-algebra has a maximal UHF subalgebra, for instance, the unital universal free product $\mathrm{M}_2 \ast_{r} \mathrm{M}_3$. As an application, we give a C*-algebraic realization of the rational subgroup $\mathbb{Q}(G,u)$ of any dimension group $G$ with order unit $u$, that is, there is a simple unital AF algebra (and a unital Kirchberg algebra) $A$ with a maximal UHF subalgebra $B$ such that $(G,u)\cong (K_0(A), [1]_0)$ and and $\mathbb{Q}(G,u)\cong K_0(B)$.