Algebra of Banach

In mathematics, the algebra of Banach is one of the fundamental structures of the analyzes functional, bearing the name of the Polish mathematician Stefan Banach (1892-1945).

Definition

A algebra of Banach $(\ mathbb has, +, \ cdot, \ times, \|\; \|)$ sur the body $\ mathbb K = \ mathbb R$ or $\ mathbb C$ is a normalized K-algebra such that the subjacent normalized vector space is moreover a Espace of Banach (a vector Space normalized complete).

According to the authors, the structure of algebra requires or not the presence of a element unit. The terms unit algebra and nonunit algebra make it possible to differentiate the structures. In functional analysis, an algebra of Banach is known as unit when there exists a neutral element, necessarily single, E , and that the standard of E is 1.

One speaks moreover about commutative algebra of Banach when the law produced is commutative.

Examples

the whole of the real numbers provided with the absolute value, the sum and the product is an algebra of Banach real and unit. In the same way, the whole of the complex numbers, provided with the module, the sum and the product is algebra of a complex and unit Banach. These examples are fundamental.

If $H$ is a real or complex vector space normalized complete, the algebra of the limited operators (i.e. real or complex endomorphisms continuous) of $H$ is an algebra of Banach real or complex (unit) for the Norme of operator S corresponding, the sum and the composition of operators. Beyond the theory results from this from the representation of the algebras of Banach.

the example (preceding) relates to in particular the algebras of endomorphisms in finished dimension: in particular, $\ mathbf {M_n (\ mathbb R)}$ and $\ mathbf {M_b (\ mathbb C)}$ are algebras of Banach, for a traditional matric standard.

the space L¹ of the integrable functions on $\ mathbb R$ (modulo the equality Almost everywhere) is a nonunit algebra of Banach relative to the Produit convolution. In the theory of Riemann of integration, this algebra is built with single isomorphism near by completion of a reasonable space, for example the space of the continuous functions with compact support, provided with the standard $\ mathbf L$ ¹.

more generally, in the theory of Lebesgue, the space of the integrable functions on a measured space is a space of Banach. The complétude request a demonstration which is based on a bringing together between L¹ convergence and convergence almost everywhere.

Properties of the unit algebras

That is to say $\ mathbb A$ a unit algebra of Banach, element unit E .

Properties of the application of passage contrary

As in any algebra, the invertible elements of has form a group. Any element X pertaining to the open ball of center E and ray 1 in fact part, and its reverse can be expressed like summons geometrical series of reason X .

\|X \|<1 \ Rightarrow (e-x) ^ {- 1} = \ sum_ {n=0} ^ {+ \ infty} x^n

The group G of the invertible elements of an algebra of Banach is a Ouvert.

The application of passage contrary is a Homéomorphisme of G on G , which confers on G a structure of topological Groupe. It is even about a differentiable application, the differential at the point X being given by the formula

d_x Inv (H) =-x^ {- 1} hx^ {- 1} \,

The assumption of complétude is essential and these results fall at fault in the noncomplete normalized algebra. For example let us consider the algebra of the polynomials with real coefficients $\ mathbb {R}$ provided with any standard. The group of invertible is $\ mathbb {R} ^ {*}$ which is included in the strict vectorial subspace $\ mathbb {R}$ of $\ mathbb {R}$ and is thus of Intérieur vacuum; it is thus not open. This shows in particular that $\ mathbb {R}$ cannot be provided with a complete structure of $\ mathbb {R}$ -algèbre normalized, result which is also consequence of the Théorème of Baire.

Ideals and algebra quotient

The maximum ideal of an algebra of Banach are closed.

An algebra of Banach complexes of which any element not no one is invertible is isomorphous, via a Isométrie, with the body of the complex numbers (Théorème of Gelfand-Mazur); in particular, the maximum ideals of the complex algebras of Banach are closed hyperplanes. To note that commutation is a consequence of the theorem.

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