Spectral Theory and Analytic Geometry Over Non-Archimedean Fields

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American Mathematical Soc., 1990 - Mathematics - 169 pages
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The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean 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 Bruhat-Tits 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 non-Archimedean 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.
 

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Contents

Introduction
1
The Spectrum of a Commutative Banach Ring
11
Affinoid Spaces
21
Analytic Spaces
47
Analytic Curves
75
Analytic Groups and Buildings
93
The Homotopy Type of Certain Analytic Spaces
115
Spectral Theory
127
Perturbation Theory
139
The Dimension of a Banach Algebra
153
References
161
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About the author (1990)

Vladimir G. Berkovich is Matthew B. Rosenhaus Professor of Mathematics at the Weizmann Institute of Science in Rehovot, Israel. He is the author of "Spectral Theory and Analytic Geometry over Non-Archimedean Fields.

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