Kernel and cokernel in the category of augmented involutive stereotype algebras

TL;DR

This paper investigates kernels and cokernels in the category of augmented involutive stereotype algebras, demonstrating their properties and implications for group algebras and duality theories.

Contribution

It establishes the existence of kernels and cokernels in this category and shows how cokernels help prove the structure of continuous envelopes as involutive Hopf algebras.

Findings

Cokernels are preserved under passage to group stereotype algebras.

The continuous envelope of certain group algebras forms an involutive Hopf algebra.

Results facilitate generalization of Pontryagin duality for Moore groups.

Abstract

We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: 1) the morphisms of the augmented involutive stereotype algebras have kernels and cokernels, 2) the cokernel is preserved under the passage to the group stereotype algebras, and 3) the notion of cokernel allows to prove that the continuous envelope $\operatorname{Env} {\mathcal C}^\star(G)$ of the group algebra ${\mathcal C}^\star(G)$ is an involutive Hopf algebra in the category of stereotype spaces $({\tt Ste},\odot)$, if $G$ has the form $Z\cdot K$, where $Z$ is a commutative locally compact group, and $K$ a compact group. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.