  We study the extension of holomorphic functions of bounded type defined on an
open subset of a Banach space, to larger domains. For this, we first
characterize the envelope of holomorphy of a Riemann domain over a Banach
space, with respect to the algebra of bounded type holomorphic functions, in
terms of the spectrum of the algebra. We then give a simple description of the
envelopes of balanced open sets and relate the concepts of domain of holomorphy
and polynomial convexity. We show that for bounded balanced sets, extensions to
the envelope are always of bounded type, and that this does not necessarily
hold for unbounded sets, answering a question posed by Hirschowitz in 1972. We
also consider extensions to open subsets of the bidual, present some
Banach-Stone type results and show some properties of the spectrum when the
domain is the unit ball of $\ell_p$.
