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.
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