Exact groups and continuous bundles of \(C^*\)-algebras.

*(English)*Zbl 0946.46054For 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 \(G\).

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 \(G\).

Reviewer: Vladimir Pestov (Wellington)

##### MSC:

46L55 | Noncommutative dynamical systems |

46L06 | Tensor products of \(C^*\)-algebras |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |