## The real rank of inductive limit $$C^*$$-algebras.(English)Zbl 0776.46025

There has lately been a renewed interest in $$C^*$$-algebras $$A$$ that are inductive limits of direct sums of $$C^*$$-algebras of the form $$C(\Omega,M_ n)$$, where $$\Omega$$ is a (connected, compact) Hausdorff space. The theory gained new momentum with G. Elliott’s classification of those algebras $$A$$ that have rank zero and stable rank one in the case where the base spaces $$\Omega$$ have a special form.
In [B. Blackadar, O. Bratteli, G. A. Elliott and A. Kumjian, Math. Ann. 292, No. 1, 111-126 (1992; Zbl 0738.46027)] it is proved that if the inductive limit $$C^*$$-algebra $$A$$ is simple and all the base spaces $$\Omega$$ are of dimension at most two, then the real rank of $$A$$ is zero if and only if projections in $$A$$ separate the traces on $$A$$. It is proved in [M. Dadarlat, G. Nagy, A. Nemethi and C. Pasnicu, Reduction of stable rank in inductive limits of $$C^*$$-algebras, Pac. J. Math. 153, 267-276 (1992; Zbl 0809.46054), see also C. R. Acad. Sci., Paris, Sér. I 312, No. 1, 107-108 (1991; Zbl 0727.46035)] that if $$A$$ is simple and the dimensions of the base spaces $$\Omega$$ are bounded, then $$A$$ has stable rank one. Combining techniques from both papers, we improve both results (the real rank result most significantly) to the case where the dimension restrictions are replaced with a weaker one: slow dimension growth.

### MSC:

 46L05 General theory of $$C^*$$-algebras

### Citations:

Zbl 0738.46027; Zbl 0727.46035; Zbl 0809.46054
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