We study permanence properties of the classes of stable and so-calledD-stable C*-algebras, respectively. More precisely, we show that aC_0(X)-algebra A is stable if all its fibres are, provided that the underlyingcompact metrizable space X has finite covering dimension or that the Cuntzsemigroup of A is almost unperforated (a condition which is automaticallysatisfied for C*-algebras absorbing the Jiang--Su algebra Z tensorially).Furthermore, we prove that if D is a K_1-injective strongly self-absorbingC*-algebra, then A absorbs D tensorially if and only if all its fibres do,again provided that X is finite-dimensional. This latter statement generalizesresults of Blanchard and Kirchberg. We also show that the condition on thedimension of X cannot be dropped. Along the way, we obtain a usefulcharacterization of when a C*-algebra with weakly unperforated Cuntz semigroupis stable, which allows us to show that stability passes to extensions ofZ-absorbing C*-algebras.
展开▼
