Dear Colleagues Here is a question. An answer results perhaps from combining Gelfand and Tannaka duality. Many thanks in advance Best wishes Johannes Let $X$ be an affine variety with coordinate ring $B$ endowed with an action $X\times G \to X$ of an affine group $G$,? and let $Y$ denote the quotient, with coordinate ring $A=B^G$. By a theorem of Oberst \cite{MR0444680}, the projection $p\colon X \to Y$ is an affine principal fiber bundle if and only if the functor from $(G,B)$-modules to $A$-modules which assigns to a? $(G,B)$-module $M$ the invariant subspace $M^G$ is an equivalence of categories. Is there a similar characterization of a topological or smooth principal bundle with structure group a (presumably compact?)? Lie group in the literature? ? @article {MR0444680, AUTHOR = {Oberst, Ulrich}, TITLE = {Affine {Q}uotientenschemata nach affinen, algebraischen {G}ruppen und induzierte {D}arstellungen}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {44}, YEAR = {1977}, NUMBER = {2}, PAGES = {503--538}, ISSN = {0021-8693}, MRCLASS = {14L99}, MRNUMBER = {0444680 (56 \#3030)}, MRREVIEWER = {T. Kambayashi}, } [For admin and other information see: http://www.mta.ca/~cat-dist/ ]