BANACH ALGEBRAS OF ULTRAMETRIC FUNCTIONS

This book examines ultrametric Banach algebras in general. It begins with algebras of continuous functions, and looks for maximal and prime ideals in connections with ultrafilters on the set of definition. The multiplicative spectrum has shown to be indispensable in ultrametric analysis and is described in the general context and then, in various cases of Banach algebras.

Applications are made to various kind of functions: uniformly continuous functions, Lipschitz functions, strictly differentiable functions, defined in a metric space. Analytic elements in an algebraically closed complete field (due to M Krasner) are recalled with most of their properties linked to T-filters and applications to their Banach algebras, and to the ultrametric holomorphic functional calculus, with applications to spectral properties. The multiplicative semi-norms of Krasner algebras are characterized by circular filters with a metric and an order that are examined.

The definition of the theory of affinoid algebras due to J Tate is recalled with all the main algebraic properties (including Krasner–Tate algebras). The existence of idempotents associated to connected components of the multiplicative spectrum is described.

Contents:

• Basic Properties in Algebra
• Norms, Semi-Norms and Multiplicative Spectrum
• Admissible 𝕃-Algebras
• Compatible Algebras
• Circular Filters and Tree Structure
• Rational Functions and Circular Filters
• Analytic Elements and T-Filters
• Holomorphic Functional Calculus
• Spectral Properties in Uniform Algebras
• Algebras Topologically of Finite Type

Readership: Graduate students and researchers in p-adic analysis, ultrametric functional analysis and ultrametric topological algebra.

Key Features:

• Order, lattices, ordered algebraic structures, Applications of Boolean algebra, Number theory, Field theory and polynomials, ultrametric fields
• Commutative rings and algebras, Sheaves, general topology, order relation, Ultrametric Banach algebras, Functional analysis