Operations on continuous bundles of \(C^*\)-algebras
We give examples of separable continuous \(C^*\)-algebra bundles \({\mathcal A}_1\), \({\mathcal A}_2\) over \(\widehat{\mathbb{N}}\) and separable \(C^*\)- algebras \(B\), \(C\) such that the fibrewise minimal \(C^*\)-algebra tensor product \({\mathcal A}_1\otimes_{\min} B\) and the fibrewise maximal \(C^*\)-algebra tensor product \({\mathcal A}_2\otimes_{\max} C\) are not continuous \(C^*\)-bundles. It follows that \({\mathcal A}_1\) is not a subbundle of a trivial bundle. We also prove the general results (i) that a \(C^*\)-algebra \(A\) is exact (respectively nuclear) if and only if for every continuous bundle of \(C^*\)-algebras the fibrewise minimal (respectively maximal) \(C^*\)-algebra tensor product with \(A\) is a continuous bundle of \(C^*\)-algebras and (ii) that a continuous \(C^*\)-algebra bundle \(\mathcal A\) with exact fibre algebras has exact bundle algebra if and only if its fibrewise minimal tensor product with any fixed \(C^*\)-algebra is continuous.
separable continuous \(C^*\)-algebra bundles
fibrewise minimal \(C^*\)- algebra tensor product
fibrewise maximal \(C^*\)-algebra tensor product
scientific article; zbMATH DE number 840206
scientific article