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

TL;DR

This paper explores the relationships between tensor products in locally convex spaces and stereotype spaces, providing identities and conditions that facilitate duality theory constructions, especially for countable discrete groups.

Contribution

It establishes identities connecting tensor products across categories and offers conditions for their equivalence, aiding duality theory development in stereotype spaces.

Findings

Identifies conditions for tensor product identities in stereotype and locally convex spaces.

Provides identities linking different tensor products and pseudosaturation operations.

Supports duality theory construction for countable discrete groups.

Abstract

We prove two identities that connect some natural tensor products in the category $\sf{LCS}$ of locally convex spaces with the tensor products in the category $\sf{Ste}$ 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 $\sf{Ste}$, $\cdot$, the primary tensor product in the category $\sf{LCS}$, and $\vartriangle$, the pseudosaturation operation in the category $\sf{LCS}$. Studying the 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 envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily, Abelian) countable discrete groups.