Gelfand theory for non-commutative Banach algebras
| Autor: | Volker Runde, El Hossein Illoussamen, Rachid Choukri |
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| Rok vydání: | 2002 |
| Předmět: |
Discrete mathematics
Pure mathematics General Mathematics 010102 general mathematics Mathematics - Operator Algebras 16. Peace & justice 01 natural sciences C*-algebra Functional Analysis (math.FA) 46H99 (primary) 46L99 Mathematics - Functional Analysis Linear map 010104 statistics & probability Banach algebra Gelfand–Naimark theorem Bijection FOS: Mathematics Homomorphism Ideal (order theory) 0101 mathematics Operator Algebras (math.OA) Commutative property Mathematics |
| DOI: | 10.48550/arxiv.math/0202306 |
| Popis: | Let $A$ be a Banach algebra. We call a pair $(G, B)$ a Gelfand theory for $A$ if the following axioms are satisfied: (G 1) $B$ is a $C^\ast$-algebra, and $G : A \to B$ is a homomorphism; (G 2) the assignment $L \mapsto G^{-1}(L)$ is a bijection between the sets of maximal modularleft ideals of $B$ and $A$, respectively; (G 3) for each maximal modular left ideal $L$ of $B$, the linear map $G_L : A / G^{-1}(L) \to B /L $ induced by $B$ has dense range. The Gelfand theory of a commutative Banach algebra is easily seen to be characterized by these axioms. Gelfand theories of arbitrary Banach algebras enjoy many of the properties of commutative Gelfand theory. We show that unital, homogeneous Banach algebras always have a Gelfand theory. For liminal $C^\ast$-algebras with discrete spectrum, we show that the identity is the only Gelfand theory (up to an appropriate notion of equivalence). Comment: 15 pages |
| Databáze: | OpenAIRE |
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