On Tensor Product α-Algebra Bundles

Abstract

This note introduces a class of bundles, called topological α-algebra bundles, where the “coefficients” of the bundles belong to a topological algebra bundle α. Furthermore, given two such bundles ξ, η, we show the existence of the tensor product algebra bundle ξ ${\mathop{ \otimes }\limits^{ \wedge }_{\alpha }}$ η and prove, under appropriate conditions, that the algebra of sections Γ(ξ ${\mathop{ \otimes }\limits^{ \wedge }_{\alpha }}$ η) is isomorphic to Γ(ξ) $\mathop{ \otimes }\limits_{{\Gamma (\alpha )}}^{ \wedge }$ Γ(η). On the other hand, given a topological algebra bundle ξ, we consider the (locally trivial fibre) “spectrum bundle” of ξ, $\mathfrak{M}$(ξ), such that the bundles ξ, $\mathfrak{M}$(ξ) are, in a sense, dual to each other. So, $\mathfrak{M}$(ξ${\mathop{ \otimes }\limits^{ \wedge }_{\alpha }}$ η) describes, under suitable conditions, the fibre product bundle of $\mathfrak{M}$(ξ), $\mathfrak{M}$(η) over $\mathfrak{M}$(α).