Maximal UHF subalgebras of certain C*-algebras

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 $\mathbb{Q}(G,u)\cong K_0(B)$.