Reduction to Dimension Three of Local Spectra of Real Rank Zero C*-algebras

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 = ⊕ 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 approximately 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. The special case of Theorem 3.4 when K(Xn,i) are torsion free was proved in [21]. The case when K(Xn,i) are torsion free is due to Gong [G]. Theorem 3.4 can be regarded as a dynamic hypostasis of Bott periodicity. This can be better understood if we compare Theorem 3.4 with a result from [19] asserting that the asymptotic homotopy type of C0(X)⊗K is determined by the K-theory group K∗(X). Here X is a compact connected metrisable space with base point and K denotes the compact operators. In particular C0(S) ⊗ K is asymptotically homotopy equivalent (but not homotopy equivalent, [24]) to C0(S )⊗K [15]. It should become now visible that the cited result is essentially a reformulation of the Bott periodicity theorem. Its proof is based on the theory of asymptotic morphisms of Connes and Higson [15] and involves a suspension theorem of [22]. In the proof of Theorem 3.2 we use the above version of Bott periodicity to replace the spaces Xn,i by lower dimensional spaces with the same K-theory groups. Due to dynamical properties of the real rank zero C*-algebras one can do these changes without changing the isomorphism class of the inductive limit C*-algebra. The present paper should be regarded as a continuation of [21]. In particular the key Lemma 1.7 which relates homotopy of approximate morphisms to approximate unitary equivalence of morphisms was proved in [21].