Structure of modules over the stereotype algebra ℒ(X) of operators

Abstract

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ℒ (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $$E \circledast X \subseteq M \subseteq E \odot X.$$ . As a corollary, we show that if X is a nuclear Frechet space with a basis, then each Frechet module M over the stereotype operator algebra ℒ(X) can be uniquely represented as the projective tensor product of X by some Frechet space E, \(M = E \widehat \otimes X\).