Decomposition rank and absorbing extensions of type I algebras
\textit{M. Pimsner, S. Popa} and \textit{D. Voiculescu} [J. Oper. Theory 1, 55--108 (1979; Zbl 0455.46060)] showed that a family of extensions satisfying the hypothesis of Voiculescu's absorption theorem at every point would, if the family satisfied a certain condition called homogeneity, be absorbing. The aim of the present paper is to generalize this result in several directions simultaneously by replacing the compact operators by a reasonably general class of type I \(C^*\)-algebras. Thus, the following absorption theorem for the case of finite decomposition rank is proved: Let \(A\) and \(B\) be separable \(C^*\)-algebras. Suppose that \(B\) is a type \(I\) \(C^*\)-algebra such that (i) \(B\) has only infinite-dimensional irreducible \(*\)-representations, and (ii) \(B\) has finite decomposition rank. If \(0\to B\to C\to A\to 0\) is a unital homogeneous exact sequence with Busby invariant \(\tau\), then the extension \(\tau\) is absorbing. Moreover, the authors give an example showing that the case of infinite decomposition rank is fundamentally different: a unital, homogeneous split exact sequence of the form \(0\to C(Z)\otimes{\mathcal K}\to C\to\mathbb{C}\to 0\) which is not absorbing is constructed, \(Z\) being an infinite-dimensional compact, second countable topological space.
scientific article; zbMATH DE number 2160153
scientific article