Multiplication operators on $L_p$ spaces and homological triviality of respective category of modules

Abstract

We give complete characterisation of topologically injective (bounded below), topologically surjective (open mapping), isometric and coisometric (quotient mapping) multiplication operators between $L_p$ spaces defined on different $\sigma$-finite measure spaces. We prove that all such operators invertible from the left or from the right. As the consequence we prove that all objects of the category of $L_p$ spaces considered as left Banach modules over algebra of bounded measurable functions are metrically, extremelly and relatively projective, injective and flat.