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

Functional Analysis and Its Applications. 2006. Vol. 40. No. 2. P. 81-90.

Akbarov S. S.

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⊛X⊆M⊆E⊙X.$E⊛X\subseteq M\subseteq E\odot X.$

. As a corollary, we show that if X is a nuclear Fréchet space with a basis, then each Fréchet module M over the stereotype operator algebra ℒ(X) can be uniquely represented as the projective tensor product of X by some Fréchet space E, M=E⊗ˆX$M=E\otimes ^X$.

Keywords: stereotype space, nuclear space, topological algebra, topological module