On tensor fractions and tensor products in the category of stereotype spaces

We prove two identities that connect some natural tensor products in the category $\operatorname{LCS}$ of locally convex spaces with tensor products in the category $\operatorname{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$X^\vartriangle\odot Y^\vartriangle\cong (X^\vartriangle\cdot Y^\vartriangle)^\vartriangle\cong (X\cdot Y)^\vartriangle$$holds, where $\odot$ is the injective tensor product in the category $\operatorname{Ste}$, $\cdot $ is the primary tensor product in $\operatorname{LCS}$, and $\vartriangle$ is the pseudosaturation operation in $\operatorname{LCS}$. The study of relations of this type is justified by the fact that they turn out to be important instruments for constructing duality theory based on the notion of an envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily Abelian) countable discrete groups. Bibliography: 15 titles.

stereotype space, pseudosaturation