Mathematical Surveys and Monographs 1990; 169 pp; softcover Volume: 33 ISBN10: 0821890204 ISBN13: 9780821890202 List Price: US$71 Member Price: US$56.80 Order Code: SURV/33.S
 The purpose of this book is to introduce a new notion of analytic space over a nonArchimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of BruhatTits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a nonArchimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and \(p\)adic analysis. Table of Contents  The spectrum of a commutative Banach ring
 Affinoid spaces
 Analytic spaces
 Analytic curves
 Analytic groups and buildings
 The homotopy type of certain analytic spaces
 Spectral theory
 Perturbation theory
 The dimension of a Banach algebra
