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.
General
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