Structure of Modules over the Stereotype Algebra $\mathcal{L}(X)$ of Operators

S. S. Akbarov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 1-12

Résumé

It is well known that every module $M$ over the algebra $\mathcal{L}(X)$ of operators on a finite-dimensional space $X$ can be represented as the tensor product of $X$ by some vector space $E$, $M\cong E\otimes X$. 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 $\mathcal{L}(X)$ 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 Fréchet space with a basis, then each Fréchet module $M$ over the stereotype operator algebra $\mathcal{L}(X)$ can be uniquely represented as the projective tensor product of $X$ by some Fréchet space $E$, $M=E\,\widehat{\otimes}\kern1pt X$.