Math. Ann. 263, 213-219 (1983) i 9 Springer-Verlag 1983 Spectrum and Envelope of Holomorphy for Infinite Dimensional Riemann Domains Martin Schottenloher Mathematisches Institut der Universit~it, Theresienstrasse 39, 8000 Miinchen 2, Federal Republic of Germany In this note we extend Rossi's result [15] on the construction of the envelope of holomorphy by means of the spectrum to domains O spread over a complex Fr6chet space E with a basis: The envelope of holomorphy g(~) of O can be identified with the space 27 of continuous characters on (0(0). Here, (_9(0) is the algebra of holomorphic functions on O with the compact open topology. This problem has previously been investigated by Alexander [1], Coeur6 [-3,4], Hirschowitz [-9], and the author [16, 17]. As a first step, Alexander endowed 27 with a natural structure of a manifold spread over E (in the case of a Banach space E). He showed that the Gelfand map 6:O--*~) is an analytic continuation of (9(Q), where ~ C 27 is that connected component of 27 containing all point evaluations 3(x):(9(O)~, x~O. In general, O is not the envelope of holomorphy of O, i.e. the analytic continuation 6 :O-~ is not maximal: Josefson [11] presented an example of a domain O in a non-separable Banach space E for which not every point of ~(O) defines a continuous character. However, Coeur6 [3] and Hirschowitz [9] proved that such characters are at least bounded. Using this result the author [16] represented g(O) as a suitably structured subset ~b of the set 2; b of all bounded characters on (_9(0). Although the problem of representing g(O) by means of the spectrum of (_9(0)seems to be settled by this result, essentially all questions concerning the non-trivial relations among the spaces ~, ~b, 27, and 27b remained unanswered. The result of this note, which is obtained by using the approach of Gruman and Kiselman [7] to the Levi problem and its consequences [18], contributes an answer to some of these questions, since it implies ~=~b = 27"-~ ~(O) whenever O is a domain spread over a Fr6chet space E with a basis. We begin with a description of the manifold structure of 2? and its basic properties. This seems to be indicated, although the construction of 27 can be found already in Alexander [1] (at least for the case of a Banach space E), since Alexander's thesis has not been published (the only accessible reference is Exbrayat [6]), and since the essential properties of 2? which we need for the proof of our result are not mentioned in [1]. Moreover, the construction of 27 can be applied to obtain results for domains spread over non-normed locally convex Hausdorff spaces [19, 21].