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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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##### References:
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