A note on Banach C0(X)-modules
摘要
The projective tensor product over C0(X) of locally C0(X)-convex non-degenerate Banach C0(X)-modules is again locally C0(X)-convex. Let X be a locally compact Hausdorff space. In this article, we discuss non-degenerate Banach C0(X)-modules, which we call C0(X)-Banach spaces. This naming is justified by the fact that C0(X)-Banach algebras are, in particular, C0(X)-Banach spaces; more precisely: a C0(X)-Banach algebra is a Banach algebra which is at the same time a C0(X)-Banach space such that the product of the algebra is compatible with the C0(X)-module structure. If E is a Banach space, then we write EX for the C0(X)-Banach space C0(X,E). The theorem about tensor products of locally C0(X)-convex spaces that we prove in this note makes it easier to compare the KKban-theories for C0(X)-Banach algebras and for upper semi-continuous fields of Banach algebras over X, see Section 1.3 of [Par07a]. We give two additional applications of the theorem at the end of the first section. In this first section, we explain what locally C0(X)-convex C0(X)-Banach spaces are. In the second section, some alternative characterisations of local C0(X)-convexity are given to facilitate the proof of the main theorem, which is carried out in the third section.