Normalized algebra

The mathematical term standardized algebra describes a certain algebraic structure on which a compatible norm is also explained.

definition

A normalized algebra is a pair consisting of an - algebra , where stands for the field of real or complex numbers, and a norm defined on , so that the following applies: ${\displaystyle (A,\|\cdot \|)}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle A}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle A}$ ${\displaystyle \|\cdot \|:A\rightarrow \mathbb {R} }$

• ${\displaystyle \|a\|=0\Leftrightarrow a=0}$ for all ${\displaystyle a\in A}$
• ${\displaystyle \|\lambda a\|=|\lambda |\cdot \|a\|}$for everyone and (homogeneity) ${\displaystyle \lambda \in \mathbb {K} }$ ${\displaystyle a\in A}$
• ${\displaystyle \|a+b\|\leq \|a\|+\|b\|}$for all ( triangle inequality ) ${\displaystyle a,b\in A}$
• ${\displaystyle \|a\cdot b\|\leq \|a\|\cdot \|b\|}$ for all ${\displaystyle a,b\in A}$

The first three norm conditions make a normalized - vector space . The last multiplicative norm condition is the condition for the multiplication, which is analogous to the additive triangle inequality, some authors therefore also speak of the multiplicative triangle inequality. This condition ensures the continuity of the multiplication, normalized algebras are therefore topological algebras . ${\displaystyle A}$ ${\displaystyle \mathbb {K} }$

Examples

• The most important examples of normalized algebras are the Banach , so those of their standard regarding fully are.
• The field with the amount as the norm is a normalized algebra. ${\displaystyle \mathbb {K} }$
• The algebra of all polynomials in an indeterminate with the norm defined by is a non-complete normalized algebra. ${\displaystyle \mathbb {K} [X]}$ ${\displaystyle \textstyle \|p\|:=\sup _{x\in [0,1]}|p(x)|}$

properties

• The standard defines a topology on the standardized algebra , the so-called standard topology . From the properties of the norm it immediately follows that the algebraic operations are continuous: Is and as well as with and , then follows , and in each case for with respect to the norm topology . ${\displaystyle A}$ ${\displaystyle a_{n}\rightarrow a}$ ${\displaystyle b_{n}\rightarrow b}$ ${\displaystyle \lambda _{n}\rightarrow \lambda }$ ${\displaystyle a_{n},a,b_{n},b\in A}$ ${\displaystyle \lambda _{n},\lambda \in \mathbb {K} }$ ${\displaystyle \lambda _{n}a_{n}\rightarrow \lambda a}$ ${\displaystyle a_{n}+b_{n}\rightarrow a+b}$ ${\displaystyle a_{n}\cdot b_{n}\rightarrow a\cdot b}$ ${\displaystyle n\to \infty }$ ${\displaystyle A}$
• Clearly, the algebraic operations continue continuously to the completion of a normalized algebra; this completion is then a Banach algebra. So every normalized algebra is densely contained in a Banach algebra.

Applications

The normalized algebras are by no means as important as the Banach algebras. However, some constructions in the theory of Banach algebras initially lead to standardized algebras, which are then completed in a subsequent construction step; Examples include the AF algebras 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 . ${\displaystyle L^{1}(G)}$

Many theorems from the theory of Banach algebras lose their validity for normalized algebras, which illuminates the importance of completeness. In the example above , the score is a discontinuous homomorphism . If a non-constant polynomial is defined as the set of all , so that it cannot be inverted, it is equal to whole , in particular not compact . Both phenomena cannot occur with Banach algebras. ${\displaystyle \mathbb {K} [X]}$ ${\displaystyle \mathbb {K} [X]\rightarrow \mathbb {K} ,\,p\mapsto p(2)}$ ${\displaystyle p\in \mathbb {K} [X]}$ ${\displaystyle \sigma _{\mathbb {K} [X]}(p)}$ ${\displaystyle \lambda \in \mathbb {K} }$ ${\displaystyle \lambda 1-p}$ ${\displaystyle \mathbb {K} }$

Local Banach algebras

For some applications, a weakened completeness property is sufficient. A normalized algebra is called a local Banach algebra if it is closed with respect to the holomorphic functional calculus . More precisely this means: Are , the spectrum formed with regard to the completion and a holomorphic function defined in a neighborhood of , with , if has no one element, then lies in . Here, in is formed according to the holomorphic functional calculus . ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle f}$ ${\displaystyle \sigma (a)}$ ${\displaystyle f(0)=0}$ ${\displaystyle A}$ ${\displaystyle f(a)}$ ${\displaystyle A}$ ${\displaystyle f(a)}$ ${\displaystyle {\overline {A}}}$

For example, if a locally compact Hausdorff space , then the algebra of all continuous functions with a compact carrier is a local Banach algebra. If it is not compact , then there is no Banach algebra. ${\displaystyle X}$ ${\displaystyle C_{c}(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X}$ ${\displaystyle C_{c}(X)}$

Deviating from this definition, inductive limits of Banach algebras are defined as local . These are apparently closed with regard to the holomorphic functional calculus, since this can be carried out in the steps of the inductive limit, which are Banach algebras.

Individual evidence

1. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862 , Chapter I. Definition 10
2. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Chapter II, 3.1
3. ↑ J. Cuntz, R. Meyer, J. Rosenberg: Topological and Bivariant K-Theory , Birkhäuser Verlag (2007), ISBN 3-764-38398-4 , definition 2.11 and the following text