  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.
