Locally AH-Algebras

Locally AH-Algebras Synopsis

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 $Pin C(X, M_n)$ is a projection, such that $ mathrm{dist}(a, B)<epsilon ext{ for all } ainmathcal {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.