Normed algebra

The mathematical term normed algebra refers to a certain algebraic structure on which an acceptable standard is also explained.

Definition

A normed algebra is a pair consisting of an algebra, where is the field of real or complex numbers, and so that the following applies to defined standard:

• For all
• For all and (homogeneity )
• For all ( triangle inequality )
• For all

Making the first three standard conditions to a normalized - vector space. The last multiplicative norm condition is analogous to the additive triangle inequality condition for multiplication, some authors therefore also speak of the multiplicative triangle inequality. This condition ensures the continuity of the multiplication normed algebras are therefore topological algebras.

Examples

• The most important examples of normed algebras are Banach algebras, ie those that are complete with respect to their standard.
• The body with the amount as the norm is a normed algebra.
• The algebra of all polynomials in one indeterminate with the norm defined by a non- complete normed algebra.

Properties

• The standard defines a topology on the normalized algebra, the so-called norm topology. From the properties of the standard results immediately that the algebraic operations are continuous: If and, with and, it follows, and each is in the norm topology.
• The algebraic operations set clearly steadily continuing on the completion of a normed algebra; this completion is then a Banach algebra. For any normed algebra is contained in a tight Banach algebra.

Applications

The normed rings have by far not as important as the Banach algebras. Some constructions in the theory of Banach algebras, however, lead initially to normalized algebras, which are then completed in a subsequent design step; examples which may be cited as the completion of inductive limits, the maximum tensor product of C *-algebras or the formation of the algebras in harmonic analysis as the completion of the corresponding algebras of continuous functions with compact support, the AF - algebras.

Many phrases from the theory of Banach algebras lose their validity for standardized algebras, which highlights the importance of completeness. In the above example, the point evaluation is a discontinuous homomorphism. Is a non- constant polynomial, so is defined as the set of all, so that it is not inverted, the same throughout, that is not particularly compact. Both phenomena can not occur in Banach algebras.

Local Banach algebras

For some applications, it comes out with a weaker completeness property. A normed algebra is called local Banach algebra if it is closed under the holomorphic functional calculus. Specifically, this means: Are the spectrum formed with respect to the completion and a function defined in a neighborhood of holomorphic function, if no one element has, as is. This is according to the holomorphic functional calculus in formed.

For example, if a locally compact Hausdorff space, then the algebra of all continuous functions with compact support is a local Banach algebra. Is not compact, so is not a Banach algebra.

Notwithstanding this definition are defined in inductive limits of Banach algebras as locally. These are obviously closed under the holomorphic functional calculus, since it can be performed in the steps of the inductive limit, who are Banach algebras.