We give several necessary
and sufficient 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.
