Locally AH-Algebras

A unital separable $C^\ast$-algebra, $A$ is said to be locally AH with no dimension growth if there is an integer $d>0$ satisfying the following: for any $\epsilon >0$ and any compact subset ${\mathcal F}\subset A,$ there is a unital $C^\ast$-subalgebra, $B$ of $A$ with the form $PC(X, M_n)P$, where $X$ is a compact metric space with covering dimension no more than $d$ and $P\in C(X, M_n)$ is a projection, such that $\mathrm{dist}(a, B)<\epsilon \text{ for all } a\in\mathcal {F}.$ The authors prove that the class of unital separable simple $C^\ast$-algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple $C^\ast$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.