An infinite-dimensional generalization of Shilov boundary and infinite-dimensional analytic structures in the spectrum of a uniform algebra

We introduce a generalization of classical Shilov boundary of a commutative Banach algebra, that is suitable for investigation of infinite-dimensional analytic structures living in the maximal ideal space (the spectrum) of a uniform algebra. The one-dimensional case, created (for the boundary) by Shilov and (for analytic structures in the spectrum) by Bishop, was carried over to n-dimensions by Sibony and Basener (for the boundary), and for n-dimensional analytic structures in the spectrum by Basener, Sibony, Kramm and others. Another, simpler definition as well as a detailed investigation of Sibony-Basener's generalization of Shilov boundary, based on the class of all nonvanishing continuous mappings from the spectrum into slash-Csup(n) has also been given. In other references this definition was carried out for the case of some classes of continuous mappings from the spectrum to a normed space. Here we interpret these results for the case of the Banach space lsup(infinity), dropping at the same time the continuity condition from the base class of mappings, connected with the corresponding infinity-generalization of Shilov boundary. In Sec. 3 an infinity-dimensional generalization of Bishop's and Basener's results about existence of analytic structure in the spectrum of a uniform algebra is given. (author)