The real rank of inductive limit \(C^*\)-algebras.

*(English)*Zbl 0776.46025There 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.

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 |