\({\mathcal{C}}_{0}\) (X)-algebras, stability and strongly self-absorbing \({\mathcal{C}}^{*}\) -algebras

摘要: We study permanence properties of the classes stable and so-called \({\mathcal{D}}\)-stable \({\mathcal{C}}^{*}\)-algebras, respectively. More precisely, we show that a \({\mathcal{C}}_{0}\) (X)-algebra A is if all its fibres are, provided underlying compact metrizable space X has finite covering dimension or Cuntz semigroup almost unperforated (a condition which automatically satisfied for \({\mathcal{C}}^{*}\)-algebras absorbing Jiang–Su algebra \({\mathcal{Z}}\) tensorially). Furthermore, prove \({\mathcal{D}}\) K 1-injective strongly self-absorbing \({\mathcal{C}}^{*}\)-algebra, then absorbs tensorially only do, again finite-dimensional. This latter statement generalizes results Blanchard Kirchberg. also on cannot be dropped. Along way, obtain useful characterization when \({\mathcal{C}}^{*}\)-algebra with weakly stable, allows us to stability passes extensions \({\mathcal{Z}}\)-absorbing \({\mathcal{C}}^{*}\) -algebras.