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On inductive limits of matrix algebras over higher dimensional spaces. I. (English) Zbl 0901.46053
Summary: Suppose that $$A$$ is a $$C^*$$-algebra of real rank zero, and is an inductive limit of $$\displaystyle{\bigoplus_{i=1}^{k_n}} M_{[n,i]} (C(X_{n,i}))$$, where the spaces $$X_{n,i}$$ are finite CW complexes, and $$[n,i]$$ are positive integers. In this note, we will prove the following results.
(1) If $$M_Q$$ is the UHF algebra with $$K_0(M_Q)= Q$$, then $$A\otimes M_Q$$ can be expressed as an inductive limit of finite direct sums of matrix algebras over $$C(S^1)$$.
(2) If one supposes further that the cohomology groups $$\widetilde{H}^* (X_{n,i})$$ are torsion free and that $$\sup_{n,i} \{\dim(X_{n,i}) \}<+\infty$$ (one can replace this condition by the condition of slow dimension growth), then $$A$$ itself can be expressed as an inductive limit of finite direct sums of matrix algebras over $$C(S^1)$$.
Recall that a result of G. Elliott says that the class of $$C^*$$-algebras of real rank zero, which can be expressed as inductive limits of finite direct sums of matrix algebras over $$C(S^1)$$, is completely classified by $$K$$-theory (graded ordered $$K$$-group with dimension range).

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L05 General theory of $$C^*$$-algebras 46M40 Inductive and projective limits in functional analysis
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