Dual algebras.
Copyright policy )Dual algebras.
Eines der grundlegenden Probleme der harmonischen Analyse lokalkompakter Gruppen besteht darin, geeignete Analoga für die Pontryagin-Dualität im nichtabelschen Fall zu finden. Hier wird man nun keine Dualität zwischen Gruppen erwarten dürfen, weil die dualen Objekte keine Gruppen mehr sind. Der Autor sucht daher nach Dualitätssätzen in der Kategorie der Banach-Algebren. Er definiert einen Dualitätsbegriff für Algebren und zeigt, daß in dem so definierten Sinne die Gruppenalgebra \(L^ 1(G)\) und die Fourier-Algebra A(G) zueinander dual sind. Dies kann als eine Dualität vom Pontryaginschen Typ angesehen werden. Der Autor zeigt ferner, daß in diesem Sinne auch für die Matrizenalgebra \(M_ n:=M(n, {\mathbb{C}})\) eine Dualitätsaussage gemacht werden kann. Und zwar sind die Algebren \((M_ n,*)\) und \((M_ n,\circ)\) zueinander dual, wobei * das gewöhnliche Matrizenprodukt und \(\circ\) das punktweise Produkt der Matrizeneinträge ist.
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\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations, dual algebras, Fourier algebra of locally compact group, duality in category of Banach algebras, Duality theorems for locally compact groups, group algebra of absolutely left Haar integrable functions, Article, Group algebras of locally compact groups, Banach algebra of completely bounded maps, 510.mathematics, Banach algebras of continuous functions, function algebras, Other transforms and operators of Fourier type, \(C^ *\)- completion, Character groups and dual objects, matrix algebras, \(L^1\)-algebras on groups, semigroups, etc.
\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations, dual algebras, Fourier algebra of locally compact group, duality in category of Banach algebras, Duality theorems for locally compact groups, group algebra of absolutely left Haar integrable functions, Article, Group algebras of locally compact groups, Banach algebra of completely bounded maps, 510.mathematics, Banach algebras of continuous functions, function algebras, Other transforms and operators of Fourier type, \(C^ *\)- completion, Character groups and dual objects, matrix algebras, \(L^1\)-algebras on groups, semigroups, etc.
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