Reduction to dimension three of of local spectra of real rank zero

In this paper we deal with C*-algebras of real rank zero that can be represented as inductive limits A = lim−→(An, νn+1,n) of direct sums of homogeneous C*-algebras of the form An = ⊕s(n)i=1 Pn,iM[n,i](C(Xn,i))Pn,i, where Xn,i are finite CW complexes and Pn,i are selfadjoint projections. Following [3] we called these C*-algebras ap-proximately homogeneous. Our main result asserts that any approximately homogeneous C*-algebra of real rank zero with supn,i(dim(Xn,i)) < ∞ or with slow dimension growth is isomorphic to an inductive limit of direct sums of homogeneous C*-algebras whose spectra have dimension at most three (see Theorem 3.2). By a remarkable recent result of Elliott and Gong, [31], the simple C*-algebras in the latter class are classified up to isomorphism by the ordered, scaled K-theory group K ∗ = K0⊕K1. Combining the two results one obtains a classification of all simple approximately homogeneous C*-algebras of real rank zero with slow dimension growth (see Theorem 3.4). The conditions on the growth of the dimensions of the local spectra Xn,i seem to be necessary in order to avoid pathologies originating in nonstable homotopy theory