Advances in ultrametric analysis : 14th International Conference on p-adic Functional Analysis, June 30 - July 4, 2016, Universite d'Auvergne, Aurillac, France / Alain Escassut, Cristina Perez-Garcia, Khodr Shamseddine editors.

Format:

Book

Conference/Event

Contributor:

Escassut, Alain, editor.

Perez-Garcia, C., 1956- editor.

Shamseddine, Khodr, 1966- editor.

Conference Name:

International Conference on p-adic Functional Analysis (14th : 2016 : Aurillac, France)

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 704

Contemporary mathematics, 704 0271-4132

Language:

English

Subjects (All):

Functional analysis--Congresses.

Functional analysis.

p-adic analysis--Congresses.

p-adic analysis.

Physical Description:

1 online resource (298 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit� d'Auvergne, Aurillac, France. Articles included in this book feature recent developments in various areas of non-Archimedean analysis: summation of p-adic series, rational maps on the projective line over \mathbb{Q}p, non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G-modules with a convex base, non-compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean K�the spaces, p-adic Nevanlinna theory and applications, and sub-coordinate representation of p-adic functions. Moreover, a paper on the history of p-adic analysis with a comparative summary of non-Archimedean fields is presented. Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area.

Contents:

Cover

Title page

Contents

Preface

Summary on non-Archimedean valued fields

Introduction.

1. Ultrametric Spaces.

- 1.1 The Strong Triangle Inequality and its Consequences.

- 1.2 Compactness and Separability.

- 1.3 Ultrametrizability.

- 1.4 Spherical Completeness.

2. Valued fields.

- 2.1 Archimedean and non-Archimedean valuations.

- 2.2 Completion Theorem.

- 2.3 Completion of Archimedean valued fields.

- 2.4 Completion of non-Archimedean valued fields.

- 2.5 Incompleteness of \QQ and ( ).

- 2.6 Compact and Locally compact valued fields.

3. Ordered fields.

- 3.1 Formally real fields.

- 3.2 General Hahn fields and the Embedding theorem.

- 3.3 Hahn Fields and Levi-Civita fields.

- 3.4 Real-closed field extensions of \RR.

4. Algebraic closure of valued fields and their completions.

- 4.1 General Results.

- 4.2 Examples.

5. General valuations.

6. Maximal, Pseudo and Spherical completeness.

- 6.1 Pseudo-completeness.

- 6.2 Spherical completeness.

- 6.3 Maximal completeness.

7. Structures of Maximally Complete fields.

- 7.1 Equal characteristic case.

- 7.2 Mixed characteristic case.

8. Catalog of fields.

9. Classification of fields.

References

Non-compact Trace class operators and Schatten-class operators in -adic Hilbert spaces

1. Introduction

2. Background

3. Schatten-class Operators in Classical Hilbert spaces

4. Schatten Class operators in -adic Hilbert spaces

5. Completely Continuous operators and Schatten-class operators

6. Trace and Schatten-class operators

Acknowledgments

Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem

2. WLUD Functions

3. Calculus Theorems

References.

adic meromorphic functions ' '( ), ' '( ) sharing a small function, ignoring multiplicity

1. Introduction and Main Results

2. Basic Results

3. Specific Lemmas

4. Proof of Theorems

Acknowledgement

Spectra of commutative non-unital Banach rings

1. Introduction, notation and preliminary

2. Double centalizer algebras and unitalizations

3. Semi-simplicity

4. Comparison with the prime spectra

5. Representations on continuous sections of bundles of Banach algebras

Ultrametric Calkin algebras

1. The completely continuous linear operators

2. Ultrametric Calkin algebras

3. Ultrametric ultraproducts of Banach spaces

4. A linear representation of the Calkin algebra when the Banach space is infinite dimensional of countable type

5. Some additional remarks

On summation of -adic series

2. Series and -adic invariant summation in integer points

3. Functional summation formula and polynomials _{ }^{ }(

)

Concluding remarks

Acknowledgements

Spectrum of ultrametric Banach algebras of strictly differentiable functions

Introduction and preliminaries

Classical -adic Nevanlinna theory and Nevalinna Theory out of a hole

I. Classical theory

II. Nevanlinna Theory out of a hole

The distance preserving mappings and isometrics defined on non-Archimedean Banach spaces

2. Results

Rational map +1/ on the projective line over ℚ_{ }

2. Preliminaries

3. Dynamical structures

Non-Archimedean Hahn-Banach theorems and injective Banach spaces

Introduction

1. Preliminaries

2. Non-Archimedean injective Banach spaces

3. Hereditary properties.

4. Some open problems and some partial answers

5. Comparison with the classical theory

-modules with a convex base

2. Basic concepts

3. Continuity of -modules with convex base

4. Maximal tight -submodules

A journey throughout the history of -adic numbers

2. How Hensel was led to create the -adic numbers

3. Definition and properties of -adic numbers

4. -adic absolute values

5. The -adic numbers at the foundation of quantum mechanics

6. Conclusion

On some classes of non-Archimedean Köthe spaces

3. Results

On the sub-coordinate representation of -adic functions

2. Sub-coordinate representation

3. Examples of the sub-coordinate representation

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4676-6