## Spectral Theory and Analytic Geometry Over Non-Archimedean FieldsThe 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 |

### Common terms and phrases

a e k a k analytic abelian variety admissible epimorphism affinoid algebras affinoid neighborhood affinoid space affinoid subdomain algebra stf analytic space assume Banach algebra Banach ring Banach space bounded homomorphism bounded linear operators canonical map closed disc closed immersion closed k-analytic closed subset coincides commutative Banach k-algebra construction COROLLARY corresponding countable defined denote dimension element embedding epimorphism equal equivalent everywhere dense fibre finite Banach finite morphisms functor Furthermore G-topology group G Hence homomorphism homotopy inverse image invertible irreducible isomorphism Jf(stf k-affinoid Lemma locally ringed spaces Max(J module multiplicative seminorm non-Archimedean field nonempty norm normal open immersion open neighborhood open subset phism point x e polynomial PROOF proper Proposition reduction resp Rigid analytic spaces Schottky group seminorm sheaf simply connected spectrum stfv subalgebra subspace suffices to show Suppose surjective Theorem topology torus trivial valuation field Weierstrass zero