Arens-Michael decomposition

The Arens-Michael decomposition , named after Richard Arens and Ernest Michael , is a mathematical construction for the investigation of LMC algebras . The Arens-Michael decomposition represents complete LMC algebras as projective limits of Banach algebras .

construction

It is an LMC-algebra, that is a topological algebra whose topology through a directed family submultiplikativer seminorms is given, and for stands. Then is a two-sided ideal and defined by ${\displaystyle A}$ ${\displaystyle (p_{\alpha })_{\alpha \in I}}$ ${\displaystyle \alpha \leq \beta }$ ${\displaystyle p_{\alpha }\geq p_{\beta }}$ ${\displaystyle N_{\alpha }:=p_{\alpha }^{-1}(0)\subset A}$ ${\displaystyle p_{\alpha }}$

${\displaystyle {\hat {p}}_{\alpha }(a+N_{\alpha }):=p_{\alpha }(a)}$

a norm on the quotient algebra . The completions of the are Banach algebras, which are denoted by. ${\displaystyle A/N_{\alpha }}$ ${\displaystyle (A/N_{\alpha },{\hat {p}}_{\alpha })}$ ${\displaystyle A_{\alpha }}$

For defines an algebra homomorphism . With these images you get an embedding ${\displaystyle \alpha \leq \beta }$ ${\displaystyle a+N_{\beta }\mapsto a+N_{\alpha }}$ ${\displaystyle f_{\alpha ,\beta }:A_{\beta }\rightarrow A_{\alpha }}$

${\displaystyle A\rightarrow \lim _{\longleftarrow }A_{\alpha }:=\{(a_{\alpha })_{\alpha };f_{\alpha ,\beta }(a_{\beta })=a_{\alpha }\,\forall \alpha <\beta \}\subset \Pi _{\alpha \in I}A_{\alpha }}$

in the projective limes of the system . Thus every LMC algebra is a subalgebra of a product of Banach algebras. This is called the Arens-Michael decomposition. ${\displaystyle (A_{\alpha },f_{\alpha ,\beta })}$

If is complete, then is surjective and the result is that complete LMC algebras are projective limits of Banach algebras. Complete LMC algebras are therefore also called Arens-Michael algebras . ${\displaystyle A}$ ${\displaystyle A\rightarrow \lim _{\longleftarrow }A_{\alpha }}$

Applications

Using the representation as projective limits of Banach algebras, some results from the theory of Banach algebras can be transferred to (complete) LMC algebras.

A typical application is Arens' invertibility criterion . With the terms from the above construction, an element from an Arens-Michael algebra with a one element can be inverted if and only if it is invertible in every algebra . ${\displaystyle a\in A}$ ${\displaystyle a+N_{\alpha }}$ ${\displaystyle A_{\alpha }}$

Furthermore, one can show with these methods that LMC algebras have a continuous inverse, that is, that the mapping on the set of invertible elements is automatically continuous. ${\displaystyle x\mapsto x^{-1}}$

Individual evidence

1. ^ EA Michael: Locally multiplicatively-convex topological algebras , Mem. Amer. Math. Soc. (1952), Volume 11
2. ↑ Anastasios Mallios: Topological Algebras: Selected Topics , North-Holland Mathematics Studies, Volume 124, Chapter III.3 "Arens Michael Decomposition"
3. ↑ AY Helemskii: The homology of Banach and Topological Algebras. Kluwer Academic Publishers (1989), ISBN 0-7923-0217-6 , Chapter 0, §1.3. Definition 1.2
4. ^ Edward Beckenstein, Lawrence Narici, Charles Suffel: Topological algebras , North-Holland Publishing Company (1977), ISBN 0-7204-0724-9 , Theorem 4.6-1 (e)
5. ^ Edward Beckenstein, Lawrence Narici, Charles Suffel: Topological algebras , North-Holland Publishing Company (1977), ISBN 0-7204-0724-9 , Theorem 4.8-6