Envelopes of Holomorphy and Extension of Functions of Bounded Type

In this article we characterize the envelope of holomorphy for the algebra of bounded type holomorphic functions on Riemann domains over a Banach space in terms of the spectrum of the algebra. We prove that evaluations at points of the envelope are always continuous but we show an example of a balanced open subset of c0 where the extensions to the envelope are not necessarily of bounded type, answering a question posed by Hirschowitz in 1972. We show that for bounded balanced sets the extensions are of bounded type. We also consider extensions to the bidual, and show some properties of the spectrum in the case of the unit ball of lp.