Extending Characters of Fixed Point Algebras

Extending Characters of Fixed Point Algebras

A dynamical system is a triple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>&#945;</mi> <mo>)</mo> </...

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Main Author: Stefan Wagner
Format: Article
Language:English
Published: MDPI AG 2018-11-01
Series:Axioms
Subjects:
dynamical system
continuous inverse algebra
character
maximal ideal
fixed point algebra
extension
Online Access:https://www.mdpi.com/2075-1680/7/4/79
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spelling doaj-5d328ff0e7ae4ac2b141e1c9c05fab8a2020-11-24T22:52:09ZengMDPI AGAxioms2075-16802018-11-01747910.3390/axioms7040079axioms7040079Extending Characters of Fixed Point AlgebrasStefan Wagner0Department of Mathematics and Natural Sciences, Blekinge Tekniska Högskola, 371 41 Karlskrona, SwedenA dynamical system is a triple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> consisting of a unital locally convex algebra <i>A</i>, a topological group <i>G</i>, and a group homomorphism <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>:</mo> <mi>G</mi> <mo>&#8594;</mo> <mo form="prefix">Aut</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> that induces a continuous action of <i>G</i> on <i>A</i>. Furthermore, a unital locally convex algebra <i>A</i> is called a continuous inverse algebra, or CIA for short, if its group of units <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> </semantics> </math> </inline-formula> is open in <i>A</i> and the inversion map <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#953;</mi> <mo>:</mo> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> <mo>&#8594;</mo> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>↦</mo> <msup> <mi>a</mi> <mrow> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> is continuous at <inline-formula> <math display="inline"> <semantics> <msub> <mn>1</mn> <mi>A</mi> </msub> </semantics> </math> </inline-formula>. Given a dynamical system <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with a complete commutative CIA <i>A</i> and a compact group <i>G</i>, we show that each character of the corresponding fixed point algebra can be extended to a character of <i>A</i>.https://www.mdpi.com/2075-1680/7/4/79dynamical systemcontinuous inverse algebracharactermaximal idealfixed point algebraextension
collection DOAJ
language English
format Article
sources DOAJ
author Stefan Wagner
spellingShingle Stefan Wagner
Extending Characters of Fixed Point Algebras
Axioms
dynamical system
continuous inverse algebra
character
maximal ideal
fixed point algebra
extension
author_facet Stefan Wagner
author_sort Stefan Wagner
title Extending Characters of Fixed Point Algebras
title_short Extending Characters of Fixed Point Algebras
title_full Extending Characters of Fixed Point Algebras
title_fullStr Extending Characters of Fixed Point Algebras
title_full_unstemmed Extending Characters of Fixed Point Algebras
title_sort extending characters of fixed point algebras
publisher MDPI AG
series Axioms
issn 2075-1680
publishDate 2018-11-01
description A dynamical system is a triple <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> consisting of a unital locally convex algebra <i>A</i>, a topological group <i>G</i>, and a group homomorphism <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>:</mo> <mi>G</mi> <mo>&#8594;</mo> <mo form="prefix">Aut</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> that induces a continuous action of <i>G</i> on <i>A</i>. Furthermore, a unital locally convex algebra <i>A</i> is called a continuous inverse algebra, or CIA for short, if its group of units <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> </semantics> </math> </inline-formula> is open in <i>A</i> and the inversion map <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#953;</mi> <mo>:</mo> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> <mo>&#8594;</mo> <msup> <mi>A</mi> <mo>&#215;</mo> </msup> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>↦</mo> <msup> <mi>a</mi> <mrow> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> is continuous at <inline-formula> <math display="inline"> <semantics> <msub> <mn>1</mn> <mi>A</mi> </msub> </semantics> </math> </inline-formula>. Given a dynamical system <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with a complete commutative CIA <i>A</i> and a compact group <i>G</i>, we show that each character of the corresponding fixed point algebra can be extended to a character of <i>A</i>.
topic dynamical system
continuous inverse algebra
character
maximal ideal
fixed point algebra
extension
url https://www.mdpi.com/2075-1680/7/4/79
work_keys_str_mv AT stefanwagner extendingcharactersoffixedpointalgebras
_version_ 1725666868296941568

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