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Exact groups and continuous bundles of $$C^*$$-algebras. (English) Zbl 0946.46054
For a locally compact group $$G$$, denote by $${\mathcal C}^\ast_G$$ the category of ‘noncommutative $$G$$-flows,’ that is, pairs $$(A,\alpha)$$ formed by a $$C^\ast$$-algebra $$A$$ and a continuous action $$\alpha$$ of $$G$$ on $$A$$, with $$G$$-equivariant $$\ast$$-homomorphisms as morphisms. The group $$G$$ is called exact if the functor from the category $${\mathcal C}^\ast_G$$ to that of $$C^\ast$$-algebras and $$\ast$$-homomorphisms, associating to a pair $$(A,\alpha)$$ as above the reduced crossed $$C^\ast$$-algebra product $$G \ltimes_{\alpha,r} A$$ preserves short exact sequences.
Let a locally compact group $$G$$ acts fibre-wise and continuously on a continuous bundle $$\mathcal A$$ of $$C^\ast$$-algebras over a locally compact Hausdorff space $$X$$. Then, if $$G$$ is exact, it follows that the reduced crossed product $$C^\ast$$-algebra bundle $$G \ltimes_{\alpha,r}{\mathcal A}$$ is continuous. One of the main results of the present article establishes the converse statement: a locally compact group $$G$$ is exact whenever the $$C^\ast$$-algebra bundle $$G \ltimes_{\alpha,r}{\mathcal A}$$ is continuous for every continuous $$C^\ast$$-algebra bundle $$\mathcal A$$ on which $$G$$ acts by a continuous field of actions.
Every amenable group $$G$$ is exact. It still remains to exhibit examples of non-exact locally compact groups (and even discrete groups). However, in the present paper it is shown that the free groups (with discrete topology) are exact.
If a locally compact group $$G$$ is exact, then the reduced group $$C^\ast$$-algebra $$C^\ast_r(G)$$ is exact. The converse statement is established in the paper for discrete $$G$$.

MSC:
 46L55 Noncommutative dynamical systems 46L06 Tensor products of $$C^*$$-algebras 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations
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