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.

P-adic Analytic Functions describes the definition and properties of p-adic analytic and meromorphic functions in a complete algebraically closed ultrametric field.Various properties of p-adic exponential-polynomials are examined, such as the Hermite-Lindemann theorem in a p-adic field, with a new proof. The order and type of growth for analytic functions are studied, in the whole field and inside an open disk. P-adic meromorphic functions are studied, not only on the whole field but also in an open disk and on the complemental of an open disk, using Motzkin meromorphic products. Finally, the p-adic Nevanlinna theory is widely explained, with various applications. Small functions are introduced with results of uniqueness for meromorphic functions. The question of whether the ring of analytic functions-in the whole field or inside an open disk-is a Bezout ring is also examined.

The book first explains the main properties of analytic functions in order to use them in the study of various problems in p-adic value distribution. Certain properties of p-adic transcendental numbers are examined such as order and type of transcendence, with problems on p-adic exponentials. Lazard's problem for analytic functions inside a disk is explained. P-adic meromorphics are studied. Sets of range uniqueness in a p-adic field are examined. The ultrametric Corona problem is studied. Injective analytic elements are characterized. The p-adic Nevanlinna theory is described and many applications are given: p-adic Hayman conjecture, Picard's values for derivatives, small functions, branched values, growth of entire functions, problems of uniqueness, URSCM and URSIM, functions of uniqueness, sharing value problems, Nevanlinna theory in characteristic p>0, p-adic Yosida's equation.

This book contains the proceedings of the 14th International Conference on $p$-adic Functional Analysis, held from June 30-July 5, 2016, at the Universite 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 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 Koethe 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.