Kernel and cokernel in the category of augmented involutive stereotype algebras

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.