Point derivations in certain sup-norm algebras

Abstract

1. Let A be a closed point-separating subalgebra of C(X) containing the constants, where X is a compact Hausdorff space. MA will denote the space of multiplicative linear functionals p on A, and to each such p we associate its kernel A,. The A, are precisely the maximal ideals of A. Under certain hypotheses, it is known that analytic discs can be embedded in MA. Wermer [W1] showed that if A is a Dirichlet algebra on X, then each Gleason part of A is either a single point or an analytic disc. Hoffman [H] then generalized Wermer's result to logmodular algebras. Finally Lumer [L] observed that the conclusion is really local: if p has a unique representing measure on X, then the part for A containing p consists either of p alone or of an analytic disc. Our objective in this paper is to take the weakest of these possible hypotheses, namely Lumer's, and show that in a broader sense the analytic disc at p, if there is one, really does account for all the analytic structure at (p. Specifically, we show that all the bounded derivations and higher derivatives of A at cp are just differentiations with respect to the analytic structure of the analytic disc.