Strong pure infiniteness of crossed products
Abstract
Consider an exact action of discrete group $G$ on a separable $C^*$-algebra $A$. It is shown that the reduced crossed product $A\rtimes_{\sigma, \lambda} G$ is strongly purely infinite - provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 4.1). This is the first non-trivial sufficient criterion for strong pure infiniteness of reduced crossed products of $C^*$-algebras $A$ that are not $G$-simple. In the case $A=\mathrm{C}_0(X)$ the notion of a $G$-separating action corresponds to the property that two compact sets $C_1$ and $C_2$, that are contained in open subsets $C_j\subseteq U_j \subseteq X$, can be mapped by elements of $g_j\in G$ onto disjoint sets $\sigma_{g_j}(C_j)\subseteq U_j$, but we do not require that $\sigma_{g_j}(U_j)\subseteq U_j$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative $C^*$-algebras $A$ (cf. Definition 6.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 6.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.5195
- arXiv:
- arXiv:1312.5195
- Bibcode:
- 2013arXiv1312.5195K
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Dynamical Systems;
- 46L35;
- 19K99;
- 46L80
- E-Print:
- 30 pages, parts were taken out and included elsewhere