Covering dimension for nuclear 𝐶*-algebras II

Winter, Wilhelm

doi:10.1090/S0002-9947-09-04602-9

Covering dimension for nuclear $C^*$-algebras II
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Trans. Amer. Math. Soc. 361 (2009), 4143-4167 Request permission

Abstract:

The completely positive rank is an analogue of the topological covering dimension, defined for nuclear $C^*$-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes.

We introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases. Furthermore, we analyze some of their properties; in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace $C^*$-algebras.

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