Banachalgebra

Banach algebras (after Stefan Banach ) are mathematical objects of functional analysis, the basis of key common properties generalize some known function spaces and operator algebras, such as spaces of continuous or integrable functions or algebra of continuous linear operators on Banach spaces.

A Banach algebra is a vector space, in which a multiplication and a standard are defined in addition that certain compatibility conditions are satisfied.

• 4.1 The identity element
• 4.2 The group of invertible elements
• 4.3 The spectrum
• 4.4 Maximum ideals

Definition

A vector space over the field or the real or complex numbers with a standard and a product is a Banach algebra if and only if:

• Is a Banach space, ie a complete normed vector space,
• Is an associative algebra,
• For all, i.e., the standard is submultiplicative.

As in the algebra commonplace the icon for the product is often omitted, only in the case of folding often the symbol, or is being used. If one demands only that there is a normed space, that is, one renounces the completeness, we obtain the more general concept of normed ring.

Special classes of Banach algebras

Banach *-algebra involutive Banach algebra or

A Banach *-algebra (over ) is a Banach algebra with an involution, so that

Some authors perpetuate the condition of isometry and then speak appropriate, by a Banach * - algebra with isometric involution. Most occur naturally involutions on Banach algebras, however, are isometric.

C * - algebras and von Neumann algebras

The Banach algebra of continuous linear operators on a Hilbert space motivates the following definition: A Banach algebra on which a semilinear involution is given in addition called C * - algebra if the so-called C * - condition is satisfied:

• For all

Such Banach algebras can be represented on Hilbert spaces. If these are then completed in a certain topology in the operator algebra on the Hilbert space, as they are called von Neumann algebras.

Examples

• Every Banach space is with the zero multiplication, that is, = 0 for all elements of the Banach space to a Banach algebra.

• Be a compact space and the space of continuous functions. With the pointwise operations and the plane defined by ( complex conjugation ) involution and the supremum is *-algebra to a commutative C. Similarly, let the space of bounded complex-valued functions on a topological space (which means the Stone - Čech compactification is equivalent) or more generally the space of C0 functions of continuous functions on a locally compact space which vanish at infinity look at.

• Be the unit circle in. It is the algebra of continuous functions that are holomorphic in the interior of D. With the pointwise operations and the plane defined by ( complex conjugation ) involution and the supremum is A (D ) to a commutative Banach * - algebra which is not a C * - algebra. This Banach algebra is also called the disc algebra.

• If V is a Banach space, then the algebra B ( V) of continuous linear operators on V is a Banach algebra which is not commutative in the case. V is a Hilbert space, then B (V ) is a - C * algebra.

• The trace class and the Hilbert-Schmidt class, or more generally, the shadow classes are examples of non-commutative Banach *-algebras that are not C * - algebras.

• In the harmonic analysis, the Banach * - algebra L1 (G ), ie, the convolution algebras over a locally compact group G is considered.

• H *-algebras are involutive Banach algebras that are both Hilbert spaces, together with an additional condition that links the involution with the Hilbert space structure.

Basics

There are some basics of the theory of Banach algebras discussed, showing an interplay between algebraic and topological properties.

The identity element

Many of the above examples are Banach algebras without a unit element. Nevertheless, such a unit element is required, one can adjoin one. In many cases it is in these approximations of the Banach algebras one; This is a topological construct which often constitutes a substitute for the lack of one element. This is especially true for C *-algebras and group algebras.

The group of invertible elements

If a Banach algebra with unit element 1, the group of invertible elements is open. If that is invertible and so is invertible, because easily considered one that converges and the inverse of is. Furthermore, the inversion is continuous as a mapping to the group of invertible elements. Therefore, a topological group.

The spectrum

In linear algebra, the amount of the eigenvalues ​​of a matrix plays an important role in the study of the matrices, that is, the elements of the Banach algebra. This generalizes the notion of spectrum:

Be a - Banach algebra with unit element. For the spectrum of, and compact by the theorem of Gelfand - Mazur not empty. For the spectral radius formula applies. This formula is amazing, since the spectral radius is a purely algebraic size, which only uses the notion of invertibility, the right side of the Spektralradiusformel other hand, is given by the norm of the Banach algebra.

For the rest of this section is commutative with identity. The set of all multiplicative functionals is called the spectrum of, or after Gelfand as Gelfand spectrum or Gelfand space of. The spectrum of is a compact space and the Gelfand transform gives a homomorphism of the Banach algebra of continuous into complex-valued functions. Each element is assigned to a continuous function, said. The spectrum of an element and the spectrum of the algebra then related by the formula. This is carried out in the article about the Gelfand transform.

Maximum ideals

Let be a commutative Banach algebra with identity -. If so is a maximal ideal ( with codimension 1). Conversely, if a maximal ideal, then the conclusion is because of the openness of the group of invertible elements of a proper ideal, the following must apply. Then the quotient algebra is a Banach algebra, which is a body, and this must be isomorphic according to the set of Gelfand - Mazur. Therefore, the quotient map is a multiplicative functional core with M. Thus, if we denote the set of maximal ideals, then one has a bijective mapping:

Thus there is a bijective relationship between the subset of the dual space and the set of maximal ideals purely algebraically defined.

Applications

• Find application Banach algebras, inter alia, in the operator theory, as used, for example, in quantum field theory.
• Furthermore, there is the extension to von Neumann algebras and Hilbert modules and abstract K- and KK - theory, which is also called non-commutative geometry.
• For the investigation of locally compact groups pulling in the harmonic analysis, the Banach algebras L1 ( G) and group C *-algebras approach.