We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact grou
p. We relate this to Tannaka duality, and to functorial properties of the group algebra. We determine the structure of the group algebra in terms of the irreducible representation, both in the real and the complex case. The particular case of a compact abelian group is worked out in detail.
A weakly complete vector space over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$ is isomorphic to $mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact
that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $expcolon Ato A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $mathbb{K}[G]$ is associated functorially so that the functor $Gmapsto mathbb{K}[G]$ is left adjoint to $Amapsto A^{-1}$. The group algebra $mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $mathcal{L}(G)$ in $A_{rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,mathbb{R})$. The dual of the group algebra $mathbb{R}[G]$ is the Hopf algebra ${mathcal R}(G,mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${mathcal R}(G,mathbb{R})leftrightarrow mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $cal K$ of spaces, and suppose that $phi:K to G$ is an abstract (i.e. not necessarily continuous) surjective group hom
omorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting.
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
