Gelfand representation of Banach modules

Kitchen, Joseph W.; Robbins, David A.

Książka - szczegóły

Tytuł książki

Gelfand representation of Banach modules

Autorzy

Joseph W. Kitchen David A. Robbins

Seria

Rozprawy Matematyczne tom/nr w serii: 203 wydano: 1982

Zawartość

Pełne teksty:

Pobierz

Warianty tytułu

Abstrakty

EN

Preface
Let A be a commutative Banach algebra with maximal ideal space ∆ and let ^: A → C₀(∆) be the Gelfand representation of A. If M is a Banach module over A, then a bounded linear map φ: M → M₀, will be called a representation of M of Gelfund type if M₀ is a Banach module over C₀(∆) and φ is ^-linear in the sense that φ(ax) = âφ(x) for all a ∈ A and x ∈ M. Two such representations have been studied previously. In [50] and [51] Robbins describes such a representation in which M₀, is taken to be a space of continuous complex-valued functions. Varela and Hofmann have considered the special case in which A is a commutative C*-algebra with identity and in that case they describe a very different type of representation in which M₀ = Γ(π),the space of continuous sections of a bundle of Banach spaces π : E → ∆. The present paper generalizes considerably the sectional representation studied by Varela and Hofmann and shows its "equivalence" to the representation studied by Robbins. For a very large class of Banach modules (M, A), including essential modules over Banach algebras with bounded approximate identities, it is shown that there is a bundle of Banach spaces π : E → ∆ and a representation ^: M → Γ₀(π) which is not only of Gelfand type, but is universal with respect to all sectional representations of Gelfand type. This representation, which is unique up to isomorphism, is called the Gelfand representation of the module. Its basic properties are developed in the second section of the paper. Flanking this section are a preliminary section concerning bundles and a section devoted to examples. The final section explores functorially the relationships between Banach modules, bundles of Banach spaces, and their morphisms.

CONTENTS
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1. Bundles of Banach spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Representations of Banach modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4. Functorial properties of the Gelfand representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Appendix. A correspondence between sections and complex-valued functions . . . . . . . . . . . . .42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Słowa kluczowe

Banach algebra Banach module bundle of Banach spaces canonical bundle section sectional representation Gelfand representation

Tematy

Kategoryzacja MSC:

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 203

Liczba stron

47

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCIII

Daty

wydano

1982

Twórcy

autor

Joseph W. Kitchen

Department of Mathematics, Duke University, Durham, U.S.A.

autor

David A. Robbins

Department of Mathematics, Trinity College, Hartford, U.S.A

Bibliografia

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Języki publikacji

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Uwagi

AMS (MO5) subject classifications. Primary, 46H15, 46H25, 55H65; Secondary, 22D15, 43A15, 43A60, 46E15, 46J20, 46M15.

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Książka - szczegóły

Gelfand representation of Banach modules

Autorzy

Seria

Zawartość

Warianty tytułu

Abstrakty

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Bibliografia

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