Banach Algebras of Ultrametric Functions

Contents
About the Author
Introduction
1. Basic Properties in Algebra
	1.1. Properties in commutative algebra
	1.2. Tree structure
	1.3. Ultrametric absolute values
2. Norms, Semi-norms and Multiplicative Spectrum
	2.1. A metric on the set of the semi-norms bounded by the norm
	2.2. L-productal vector spaces
	2.3. Multiplicative semi-norms
	2.4. Shilov boundary
	2.5. Spectral semi-norm
	2.6. Topological divisors of zero
3. Admissible L-Algebras
	3.1. Definition of admissible L-algebras
	3.2. Sticked ultrafilters
	3.3. Maximal and prime ideals of S
	3.4. Maximal ideals of finite codimension for admissible algebras
	3.5. Multiplicative spectrum of admissible algebras
	3.6. The Stone space IB(E) for admissible algebras
4. Compatible Algebras
	4.1. Definition of compatible algebras
	4.2. Maximal and prime ideals of T
	4.3. Multiplicative spectrum of T
	4.4. The Stone space of G(E)
	4.5. About the completion of E
	4.6. Algebras B, L, D, E
	4.7. Particular properties of the algebras B, L, D, E
	4.8. A kind of Gelfand transform
5. Circular Filters and Tree Structure
	5.1. Infraconnected sets
	5.2. Monotonous filters
	5.3. Circular filters
	5.4. Tree structure and metric on circular filters
6. Rational Functions and Circular Filters
	6.1. Rational functions and algebras R(D)
	6.2. Pointwise topology on Mult(K[x])
	6.3. Topologies on Mult(K[x])
7. Analytic Elements and T-Filters
	7.1. Analytic elements
	7.2. Properties of the function Ψ for analytic elements
	7.3. Holomorphic properties on infraconnected sets
	7.4. T-filters and T-sequences
	7.5. Examples and counter-examples
	7.6. Characteristic property of T-filters
	7.7. Applications of T-filters
	7.8. The p-adic Fourier transform
8. Holomorphic Functional Calculus
	8.1. Analytic elements on classic partitions
	8.2. Holomorphic properties on partitions
	8.3. Shilov boundary for algebras (H(D), O)
	8.4. Holomorphic functional calculus
9. Spectral Properties in Uniform Algebras
	9.1. Spectral properties
	9.2. Uniform K-Banach algebras and properties (s) and (q)
	9.3. Properties (o) and (q) in uniform Banach K-algebras
	9.4. Properties (o) and (q) and strongly valued fields
	9.5. Multbijective Banach K-algebras
	9.6. Polnorm on algebras and algebraic extensions
10. Algebras Topologically of Finite Type
	10.1. Hensel lemma
	10.2. Definitions of affinoid algebras
	10.3. Salmon’s theorem
	10.4. Algebraic properties of affinoid algebras
	10.5. Jacobson radical of affinoid algebras
	10.6. Spectral norm of affinoid algebras
	10.7. Spectrum of an element of an affinoid algebra
	10.8. Topologically separable fields
	10.9. Krasner–Tate algebras
	10.10. Associated idempotents
Bibliography
Definitions and Notations
Index