Abstract

We give several necessary and suficient conditions for an AH algebra to have its ideals generated by their projections. Denote by C the class of AH algebras as above and in addition with slow dimension growth. We completely classify the algebras in C up to a shape equivalence by a K-theoretical invariant. For this, we show first, in particular, that any C^*-algebra in C is shape equivalent to an AH algebra with slow dimension growth and real rank zero (generalizing so a result of Elliott-Gong); then, we use a classification result of Dadarlat-Gong. We prove that any AH algebra in C has stable rank one (i.e., in the unital case, that the set of the invertible elements is dense in the algebra), generalizing results of Blackadar-Dadarlat-Rørdam and of Elliott-Gong. Other nonstable K-theoretical results for C^*-algebras in C are also proved, generalizing results of Dadarlat-Némethi, Martin-Pasnicu and Blackadar.

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