Tensor products and multipliers of modules L p on locally compact measure spaces

Abstract

Projective module tensor products and spaces of multipliers (i.e., bounded module morphisms) of the spaces Lp(µ) and Lq(ν) regarded as modules over the algebras C0(Ω) and B(Ω) on a locally compact space Ω are described. Here B(Ω) consists of bounded Borel functions on Ω, µ and ν are regular Borel measures on Ω, 1 ≤ p, q ≤ ∞ in the case of the base algebra B(Ω), and 1 ≤ p, q < ∞ in the case of the base algebra C0(Ω). (Loosely speaking, both the tensor product and the space of multipliers turn out to be yet other modules, which consist of integrable functions and correspond to their own subscripts on L and measures). It is proved and used as an auxiliary tool that, in the case p, q < ∞ (and, generally, only in this case), the replacement of the base algebra C0(Ω) by B(Ω) leaves the tensor products and multipliers intact.