A note on Banach C0(X)-modules

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. A Banach C0(X)-module is a Banach space E which is at the same time a (left) C0(X)-module such that ‖χ · e‖ ≤ ‖χ‖∞ ‖e‖ for all χ ∈ C0(X) and e ∈ E; such an E is called non-degenerate if C0(X)E is dense in E. We suggest to call a non-degenerate Banach C0(X)-module a C0(X)-Banach space. 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 KK-theories for C0(X)-Banach algebras and for upper semi-continuous fields of Banach algebras over X , see Section 1.3 of [11]. 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. The results of this note are contained in the doctoral thesis [12]; I thank my supervisor Siegfried Echterhoff for his numerous suggestions for improvement. This research has been supported by the Deutsche Forschungsgemeinschaft (SFB 478). 1. LOCALLY C0(X)-CONVEX C0(X)-BANACH SPACES The notion of local C0(X)-convexity is crucial if one wants to compare the concept of a C0(X)-Banach space to the concepts of bundles or sheaves or (upper semicontinuous) fields of Banach spaces over X (see [6]). The same notion appears under different names in the literature (for compact X , say): Hofmann calls it “local C(X)convexity” in [? ] and so does Gierz in his extensive discussion [4]; the same name is used in [6], but there also the abbreviated form “local convexity” is proposed. Dupre and Gillette use “C(X)-convexity” in [2]. Kitchen and Robbins prefer “C(X)-local 100000 WALTHER PARAVICINI convexity” in [8] which is in line with similar concepts in [10]. Finally, the same notion appears as “regularity” of C0(X)-Banach algebras in the preprint [9]. For now, we stick to the lengthy but accurate name “local C(X)-convexity”: Definition 1.1. A C0(X)-Banach space is called locally C0(X)-convex if ‖χ1e1 + χ2e2‖ ≤ max{‖e1‖ , ‖e2‖} holds for all χ1, χ2 ∈ C0(X) with χ1, χ2 ≥ 0 and χ1 +χ2 ≤ 1 and for all e1, e2 ∈ E. Closed subspaces, quotients and finite products of locally C0(X)-convex C0(X)Banach spaces are again locally C0(X)-convex. These and other properties of the category of locally C0(X)-convex C0(X)-Banach spaces were studied in [8]. In the present article, we show the following additional result: Theorem 1.2. Let E and F be locally C0(X)-convex C0(X)-Banach spaces. Then their C0(X)-tensor product E ⊗C0(X) F is locally C0(X)-convex. By C0(X)-tensor product we mean the following: Definition 1.3. Let E and F be C0(X)-Banach spaces. Then the projective tensor product over C0(X) or C0(X)-tensor productE⊗C0(X)F ofE and F is defined to be the quotient of the complete projective tensor product E⊗ F by the closed subspace generated by elements of the form χe ⊗ f − e ⊗ χf , where χ ∈ C0(X), e ∈ E and f ∈ F . Note that the C0(X)-tensor product has the universal property for C0(X)-balanced continuous bilinear maps on E × F ; a bilinear map β from E × F to some Banach space G is called C0(X)-balanced if β(χe, f) = β(e, χf) for all χ ∈ C0(X), e ∈ E and f ∈ F . In [8], the tensor product of locally C0(X)-convex C0(X)-Banach spaces was defined to be the so-called Gelfand transform of our C0(X)-tensor product to make sure that it was also locally C0(X)-convex. Theorem 1.2 shows that this extra step is not necessary. One reason to consider locally C0(X)-convex C0(X)-Banach spaces is that they are determined by their fibres: If E is a C0(X)-Banach space, then the subspace C0(U)E is closed in E for every open subset U ⊆ X . If x ∈ X , then the fibre Ex of E is defined as the quotient Ex := E / (C0(X \ {x})E) . For all e ∈ E, we will denote by ex the corresponding element of the fibre Ex. The canonical projection map from E onto Ex will be denoted by π x . For all e ∈ E, the function x 7→ ‖ex‖Ex is upper semi-continuous and vanishes at infinity. For example, if E = E0X = C0(X,E0) for some Banach space E0, then Ex ∼= E0 for all x ∈ X and the map π x can be identified with evaluation at x. In [2], Theorem 2.5. , it is shown that E is locally C0(X)-convex if and only if ‖e‖ = sup x∈X ‖ex‖ for all e ∈ E, i.e., the Gelfand representation in the sense of [8] is isometric. Munster Journal of Mathematics VOL. 1 (2008), 99999–99999 A NOTE ON BANACH C0(X)-MODULES 100001 There is an important consequence of local C0(X)-convexity which concerns linear operators between C0(X)-Banach spaces. Note that C0(X)-Banach spaces form a category: If E and F are C0(X)-Banach spaces, then we take the bounded linear C0(X)-linear maps from E to F as morphisms and denote the set of these by LC0(X)(E,F ). Let E and F be C0(X)-Banach spaces and let x ∈ X . Let T ∈ LC0(X) (E,F ). Then there is a unique linear map Tx : Ex → Fx such that the following diagram commutes