Group Algebra

Version	Summary	Created by	Modification	Content Size	Created at	Operation
1	handwiki	Rita Xu	--	2919	2022-11-02 01:34:18

The content is sourced from: https://handwiki.org/wiki/Group_algebra

In functional analysis and related areas of mathematics, group algebras are constructions that generalize the concept of group ring to some classes of topological groups with the aim to reduce the theory of representations of topological groups to the theory of representations of topological algebras. There are several nonequivalent definitions of group algebra, each of which is considered convenient in a particular situation.

functional analysis topological groups group algebras

References

Lang 2002, Chapter XVIII.
Vinberg 2003, 12.4.
Akbarov 2003, p. 272.
If [math]\displaystyle{ G }[/math] is an infinite locally compact group then the algebra [math]\displaystyle{ {\mathcal C}^\star(G) }[/math] of measures on [math]\displaystyle{ G }[/math] is not a Fréchet algebra. In the case when [math]\displaystyle{ G }[/math] is compact, [math]\displaystyle{ {\mathcal C}^\star(G) }[/math] is a Smith space. If [math]\displaystyle{ G }[/math] is [math]\displaystyle{ \sigma }[/math]-compact, then [math]\displaystyle{ {\mathcal C}^\star(G) }[/math] is a Brauner space.
A Stein group is a complex Lie group [math]\displaystyle{ G }[/math] which is a Stein manifold.
We use here the definition given above: a map [math]\displaystyle{ \pi:G\to A }[/math] of a group [math]\displaystyle{ G }[/math] into a unital associative algebra [math]\displaystyle{ A }[/math] over [math]\displaystyle{ K }[/math] is called a representation of the group [math]\displaystyle{ G }[/math] in the algebra [math]\displaystyle{ A }[/math], if it preserves the unit and the multiplication: [math]\displaystyle{ \pi(1_G)=1_A,\quad \pi(s\cdot t)=\pi(s)\cdot\pi(t),\quad s,t\in G. }[/math]
Akbarov 2003, p. 275.
A map [math]\displaystyle{ \xi:M\to X }[/math] of a smooth manifold [math]\displaystyle{ M }[/math] into a stereotype space [math]\displaystyle{ X }[/math] is said to be smooth if for each functional [math]\displaystyle{ f\in X^\star }[/math] the composition [math]\displaystyle{ f\circ\xi }[/math] is a smooth function on [math]\displaystyle{ M }[/math], and the map [math]\displaystyle{ f\in X^\star\mapsto f\circ\xi\in {\mathcal E}(M) }[/math] is continuous.
A map [math]\displaystyle{ \xi:M\to X }[/math] of a Stein manifold [math]\displaystyle{ M }[/math] into a stereotype space [math]\displaystyle{ X }[/math] is said to be holomorphic if for each functional [math]\displaystyle{ f\in X^\star }[/math] the composition [math]\displaystyle{ f\circ\xi }[/math] is a holomorphic function on [math]\displaystyle{ M }[/math], and the map [math]\displaystyle{ f\in X^\star\mapsto f\circ\xi\in {\mathcal O}(M) }[/math] is continuous.
A map [math]\displaystyle{ \xi:M\to X }[/math] of an affine algebraic variety [math]\displaystyle{ M }[/math] over [math]\displaystyle{ {\mathbb C} }[/math] into a stereotype space [math]\displaystyle{ X }[/math] is said to be polynomial (or regular) if for each functional [math]\displaystyle{ f\in X^\star }[/math] the composition [math]\displaystyle{ f\circ\xi }[/math] is a polynomial on [math]\displaystyle{ M }[/math], and the map [math]\displaystyle{ f\in X^\star\mapsto f\circ\xi\in {\mathcal P}(M) }[/math] is continuous.
Akbarov 2009, p. 507.

©Text is available under the terms and conditions of the Creative Commons-Attribution ShareAlike (CC BY-SA) license; additional terms may apply. By using this site, you agree to the Terms and Conditions and Privacy Policy.

Upload a video for this entry

Information

Subjects: Others

Contributor MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register :

HandWiki

View Times: 1.9K

Entry Collection: HandWiki

Update Date: 02 Nov 2022

Table of Contents

Video Upload Options

References