$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras
$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras
复制标题
$${mathcal{C}}_{0}$$ (X)-代数，稳定性和强自吸收 $${mathcal{C}}^{*}$$ -代数
DOI：
10.1007/s00208-007-0129-8
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发表时间：
2007
影响因子：
1.4
通讯作者：
W. Winter
中科院分区：
文献类型：
--
作者：
Ilan Hirshberg;M. Rørdam;W. Winter
We study permanence properties of the classes of stable and so-called $${\mathcal{D}}$$-stable $${\mathcal{C}}^{*}$$-algebras, respectively. More precisely, we show that a $${\mathcal{C}}_{0}$$ (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for $${\mathcal{C}}^{*}$$-algebras absorbing the Jiang–Su algebra $${\mathcal{Z}}$$ tensorially). Furthermore, we prove that if $${\mathcal{D}}$$ is a K1-injective strongly self-absorbing $${\mathcal{C}}^{*}$$-algebra, then A absorbs $${\mathcal{D}}$$ tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a $${\mathcal{C}}^{*}$$-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of $${\mathcal{Z}}$$-absorbing $${\mathcal{C}}^{*}$$ -algebras.
We study permanence properties of the classes of stable and so-called $${\mathcal{D}}$$-stable $${\mathcal{C}}^{*}$$-algebras, respectively. More precisely, we show that a $${\mathcal{C}}_{0}$$ (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for $${\mathcal{C}}^{*}$$-algebras absorbing the Jiang–Su algebra $${\mathcal{Z}}$$ tensorially). Furthermore, we prove that if $${\mathcal{D}}$$ is a K1-injective strongly self-absorbing $${\mathcal{C}}^{*}$$-algebra, then A absorbs $${\mathcal{D}}$$ tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a $${\mathcal{C}}^{*}$$-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of $${\mathcal{Z}}$$-absorbing $${\mathcal{C}}^{*}$$ -algebras.