Locally Ah-algebras

A unital separable $Cast$-algebra, $A$ is said to be locally AH with no dimension growth if there is an integer $d0$ satisfying the for any $epsilon 0$ and any compact subset $mathcal Fsubset A,$ there is a unital $Cast$-subalgebra, $B$ of $A$ with the form $PC(X, Mn)P$, where $X$ is a compact metric space with covering dimension no more than $d$ and $Pin C(X, Mn)$ is a projection, such that $mathrmdist(a, B)The authors prove that the class of unital separable simple $Cast$-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 $Cast$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.