Vol. 94, No. 1, 1981

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Tensor products of Banach bundles

Joseph Weston Kitchen, Jr. and David A. Robbins

Vol. 94 (1981), No. 1, 151–169

This paper is concerned with projective and inductive tensor products of bundles of Banach spaces. Let π : E S and ρ : F T be bundles of Banach spaces over the locally compact Hausdorff spaces S and T, with fibers {Es : s S} and {Ft : t T}, respectively. Let Γ0(π) and Γ0(ρ) be their spaces of sections which disappear at infinity. We show the existence of a bundle πρ : EF S ×T whose fibers are {EsEt : (s,t) S ×T}; if σ Γ0(π) and τ Γ0(ρ), then their pointwise tensor σ τ defined by (σ τ)(s,t) = σ(s) τ(t) is a section of the bundle πρ : EF S × T. Further, we show the existence of a bundle πρ : EF S × T whose fibers are {EsFt : (s,t) S × T}, and demonstrate that Γ0(π)Γ0(ρ) and Γ0(πρ) are isometrically isomorphic.

Mathematical Subject Classification 2000
Primary: 46M05
Secondary: 46H25, 46M15
Received: 24 October 1979
Revised: 30 July 1980
Published: 1 May 1981
Joseph Weston Kitchen, Jr.
David A. Robbins