دانلود کتاب جبر باناخ از توابع اولترا متریک

Let L be a complete ultrametric field and let A be a Banach L-algebra of functions defined in an ultrametric space E, with values in L, satisfying certain properties (bounded continuous functions, Lipschitz functions, differentiable functions, analytic functions of several kinds). We examine algebraic properties and topologic properties they satisfy, gathering various results obtained during the last 50 years. Many properties are linked to the multiplicative spectrum of the algebras. Admissible algebras (denoted by S) are Banach algebras of functions on an ultrametric space E whose spectral semi-norm is the norm of uniform convergence, where each closed open subset has a characteristic function and any function is invertible if and only if its minimum is strictly positive. Particularly, the algebra of bounded continuous function from E to L is admissible. We show the role of sticked ultrafilters and obtain an equivalence relation whose classes characterize the maximal ideals. A prime ideal is contained in a unique maximal ideal. A prime ideal of S is a maximal ideal if and only if it equals its closure with respect to the uniform convergence. That notion particularly applies to continuous functions. Two main topologies are defined on an algebra S: the classical topology and the spectral topology defined by the spectral semi-norm. Each ultrafilter on E defines a maximal ideal of the algebra S and two ultrafilters define the same maximal ideal if and only if they are sticked. On an admissible algebra S, a kind of Bezout– Corona theorem works and that shows the role of filters concerning all ideals of S. Given a prime ideal, its closure with respect to the classical topology of S is a maximal ideal. If the field L is perfect, then all maximal ideals of S are of codimension 1. Every continuous multiplicative semi-norm of an algebra S has a kernel that is a maximal ideal and each maximal ideal of S is the kernel of a unique continuous multiplicative semi-norm (which is not true in certain ultrametric Banach algebras, details in what follows). The multiplicative spectrum appears as a compactification of the space E and the Shilov boundary of S is equal to the multiplicative spectrum of S. The stone space St(E) is defined as usual and appears as a compactification of E that actually, is homeomorphic to the multiplicative spectrum.