Topological algebra

A topological algebra is a mathematical structure. It is an algebra , usually over the field of real or complex numbers, that carries a topology so that the algebraic operations, i.e. addition, multiplication and scalar multiplication, are continuous . Such algebras, the most prominent representatives of which are Banach algebras , are examined in functional analysis. ${\displaystyle \mathbb {K} }$

definition

A topological algebra ${\displaystyle \mathbb {K} }$ is an algebra such that the mappings ${\displaystyle \mathbb {K} }$ ${\displaystyle A}$

• ${\displaystyle A\times A\rightarrow A,\,(a,b)\mapsto a+b}$
• ${\displaystyle A\times A\rightarrow A,\,(a,b)\mapsto a\cdot b}$
• ${\displaystyle \mathbb {K} \times A\rightarrow A,\,(\lambda ,a)\mapsto \lambda a}$

are steady. is thus a topological vector space on which a continuous multiplication is defined. ${\displaystyle A}$

Important classes

Banach algebras

→ Main article : Banach algebra

The best known examples are normalized algebras , especially Banach algebras . A comprehensive theory has been developed for the latter in particular. Important special cases are C * algebras , especially Von Neumann algebras , and group algebras in harmonic analysis . ${\displaystyle L^{1}(G)}$

Fréchet algebras

These are algebras that form a Fréchet space with respect to a sequence of submultiplicative semi - norms . The sub-multiplicativity of the semi-norms ensures the continuity of the multiplication. ${\displaystyle (p_{n})_{n}}$

The algebra of all continuous functions on a separable , locally compact Hausdorff space becomes a Fréchet algebra if the topology is defined by the semi-norms ${\displaystyle \mathbb {C} }$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X}$

${\displaystyle \textstyle p_{n}(f):=\sup _{x\in K_{n}}|f(x)|}$

defined, where is a sequence of compact sets for which lies in the interior of and which satisfy. then carries the topology of compact convergence and is therefore also referred to as. ${\displaystyle K_{n}}$ ${\displaystyle K_{n}\subset X}$ ${\displaystyle K_{n}}$ ${\displaystyle K_{n+1}}$ ${\displaystyle \textstyle X=\bigcup _{n\in \mathbb {N} }K_{n}}$ ${\displaystyle C(X)}$ ${\displaystyle C_{c}(X)}$

If especially an open set, then the algebra of the holomorphic functions forms a sub-Fréchet-algebra of . These algebras cannot be normalized , in particular no Banach algebras, they play a role in the function theory of several variables . ${\displaystyle X\subset \mathbb {C} ^{n}}$ ${\displaystyle H(X)}$ ${\displaystyle C_{c}(X)}$

LMC algebras

An LMC algebra, or locally multiplicative-convex algebra, is an algebra with a locally convex topology that is defined by a family of submultiplicative semi-norms. The submultiplicativity ensures the continuity of the multiplication. The complete LMC algebras are also called Arens-Michael algebras , they can be examined using the Arens-Michael decomposition .

Let be a topological space and the -algebra of continuous functions with the topology of point-wise convergence. This is defined by the family of submultiplicative semi-norms , where . If it is uncountable , then there is no Fréchet algebra. ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle X\rightarrow \mathbb {K} }$ ${\displaystyle p_{x}(f):=|f(x)|}$ ${\displaystyle x\in X}$ ${\displaystyle X}$ ${\displaystyle C(X)}$

Local convex algebras

A topological algebra is called locally convex algebra if its topology is locally convex. By definition, LMC algebras are locally convex, but the topology of a locally convex algebra is not necessarily generated by a family of submultiplicative semi-norms.

As an example we consider algebra , the quotient field of the polynomial ring . We define for functions ${\displaystyle \mathbb {C} (t)}$ ${\displaystyle \mathbb {C} [t]}$ ${\displaystyle n\geq 1}$

${\displaystyle w_{n}:\mathbb {Z} \rightarrow \mathbb {R} ^{+},\,w_{n}(k)={\begin{cases}(1-k)^{n(1-k)}&{\text{falls }}k\leq -1\\1&{\text{falls }}k=0\\(1+k)^{-(k+1)/n}&{\text{falls }}k\geq 1\end{cases}}}$

Each element can be understood as a function of a complex variable and as such has a Laurent expansion . Now define the semi-norm on by ${\displaystyle f\in \mathbb {C} (t)}$ ${\displaystyle \textstyle f(t)=\sum _{k=-\infty }^{\infty }a_{k}t^{k}}$ ${\displaystyle p_{n}}$ ${\displaystyle \mathbb {C} (t)}$

${\displaystyle \textstyle p_{n}(f):=\sum _{k=-\infty }^{\infty }|a_{k}|w_{n}(k),\quad {\mbox{falls}}\quad f(t)=\sum _{k=-\infty }^{\infty }a_{k}t^{k}}$.

One can show that with the semi-norms there is a locally convex algebra that is not an LMC algebra. ${\displaystyle \mathbb {C} (t)}$ ${\displaystyle (p_{n})_{n}}$

properties

Important properties of Banach algebras do not apply to more general classes. The automatic continuity of homomorphisms from the algebra to the basic field, which is given in Banach algebras, is an open problem in Fréchet algebras. Other typical properties of Banach algebras are also required in more general situations. This then leads to further classes of algebras.

Q algebras

A topological algebra with one element is called Q-algebra if the set of invertible elements is open. A topological algebra with one element is a Q-algebra if and only if the interior of is not empty. The spectrum of an element of a Q-algebra, that is, the set , is compact . ${\displaystyle A}$ ${\displaystyle A^{-1}}$ ${\displaystyle A^{-1}}$ ${\displaystyle a}$ ${\displaystyle \{\lambda \in \mathbb {C} ;\,\lambda \cdot 1-a\notin A^{-1}\}\subset \mathbb {C} }$

Every Banach algebra is a Q algebra, the Fréchet algebra is not a Q algebra. ${\displaystyle C_{c}(\mathbb {R} )}$

Algebras with continuous inverses

If the mapping is continuous in a topological algebra with one element , it is said to be an algebra with continuous inverses. The above example of a locally convex algebra has no continuous inverses. Using the Arens-Michael decomposition one can show that LMC algebras have continuous inverses. ${\displaystyle A}$ ${\displaystyle A^{-1}\rightarrow A^{-1},\,x\rightarrow x^{-1}}$ ${\displaystyle A}$ ${\displaystyle \mathbb {C} (t)}$

literature

• Edward Beckenstein, Lawrence Narici, Charles Suffel: Topological algebras , North-Holland Publishing Company (1977), ISBN 0-7204-0724-9