Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces

Abstract

Weakly complete real or complex associative algebras A are necessarily projective limits of finite dimensional algebras. Their group of units A−1 is a pro-Lie group with the associated topological Lie algebra ALie of A as Lie algebra and the globally defined exponential function exp:A→A−1 as the exponential function of A−1. With each topological group G, a weakly complete group algebra K[G] is associated functorially so that the functor G↦K[G] is left adjoint to A↦A−1. The group algebra K[G] is a weakly complete Hopf algebra. If G is compact, then R[G] contains G as the set of grouplike elements. The category of all real weakly complete Hopf algebras A with a compact group of grouplike elements whose linear span is dense in A is equivalent to the category of compact groups. The group algebra A=R[G] of a compact group G contains a copy of the Lie algebra L(G) in ALie; it also contains all probability measures on G. The dual of the group algebra R[G] is the Hopf algebra R(G,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 R(G,R)↔R[G] and thus yields a new aspect of Tannaka duality. In the case of a compact abelian G, an alternative concrete construction of K[G] is given both for K=C and K=R. Because of the presence of L(G), the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with K[G].