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Gelfand representation of Banach modules

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Kitchen J. W., Robbins D. A.

Instytut Matematyczny Polskiej Akademii Nauk

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

Ograniczanie wyników

1 Kitchen J. W.

1 Robbins D. A.

1 1982

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Gelfand representation of Banach modules