Tensor Products and Multipliers of Modules $L_p$ on Locally Compact Measure Spaces

Projective module tensor products and spaces of multipliers (i.e., bounded module morphisms) of the spaces $L_p(\mu)$ and $L_q(\nu)$ regarded as modules over the algebras $C_0(\Omega)$ and $B(\Omega)$ on a locally compact space $\Omega$ are described. Here $B(\Omega)$ consists of bounded Borel functions on $\Omega$, $\mu$ and $\nu$ are regular Borel measures on $\Omega$, $1\le p,q\le\infty$ in the case of the base algebra $B(\Omega)$, and $1\le p,q<\infty$ in the case of the base algebra $C_0(\Omega)$. (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 $p,q<\infty$ (and, generally, only in this case), the replacement of the base algebra $C_0(\Omega)$ by $B(\Omega)$ leaves the tensor products and multipliers intact.