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Reduction of real rank in inductive limits of \(C^*\)-algebras. (English) Zbl 0738.46027
We consider inductive limits \({\mathfrak A}\) of sequences \({\mathfrak A}_ 1\to {\mathfrak A}_ 2\to\cdots\) of finite direct sums of \(C^*\)-algebras of continuous functions from connected compact Hausdorff spaces into full matrix algebras. First we study general conditions ensuring that \({\mathfrak A}\) has real rank zero and prove for example that if \({\mathfrak A}\) is simple and the spectra of the \({\mathfrak A}_ n\) are at most two- dimensional, then \({\mathfrak A}\) has real rank zero if and only if the projections of \({\mathfrak A}\) separate the tracial states on \({\mathfrak A}\). Next we specialize to the case that each \({\mathfrak A}_ n\) is the algebra of continuous functions from the circle into a full matrix algebra, and the embeddings of \({\mathfrak A}_ n\) into \({\mathfrak A}_{n+1}\) are direct sums of standard wound embeddings in the sense of B. Blackadar [Ann. Math., II. Ser. 131, 589-623 (1990; Zbl 0718.46024)]. We prove that \({\mathfrak A}\) has real rank zero if and only if the number of standard \(\pm 1\)-times around embeddings is asymptotically small compared with the total number of embeddings as \(n\to \infty\). The latter property is expressed precisely by the absence of certain atoms for an infinite product measure associated to \(\mathfrak A\). Since \(\mathfrak A\) is simple unless all embeddings are direct sums of \(\pm 1\)-times around embeddings for large enough \(n\), this gives a counterexample to conjecture 6.1.5 in the preprint of Blackadar’s paper cited above.
Reviewer: O.Bratteli (Oslo)

MSC:
46L05 General theory of \(C^*\)-algebras
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