Point derivations and analytic structure in the spectrum of a Banach algebra

AbstractLet A be a commutative complex Banach algebra with unit. Via the Gelfand map, ƒ → ƒ, A may be represented as an algebra of continuous functions on Spec A, the spectrum (maximal ideal space) of A. It is shown that if the space of point derivations on A at θ ϵ Spec A has finite dimension n, then there exists an analytic variety V in a polycylinder in Cn and a homeomorphism τ of V onto a neighborhood of θ (in the metric topology which Spec A inherits from A∗), such that ƒ ○ τ is analytic on V for every ƒ in A, This generalizes a theorem of Gleason