The Banach algebra of continuous bounded functions with separable support

• Autorzy: M. R. Koushesh
• Języki publikacji: EN

Abstrakty

EN

We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s}(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable, is moreover non-normal; in addition C₀(Y) = C₀₀(Y). When the underlying field of scalars is the complex numbers, the space Y coincides with the spectrum of the C*-algebra $C_{s}(X)$. Further, we find the dimension of the algebra $C_{s}(X)$.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46J10: Banach algebras of continuous functions, function algebras
• 16S60: Rings of functions, subdirect products, sheaves of rings
• 54C35: Function spaces
• 46J25: Representations of commutative topological algebras
• 46E25: Rings and algebras of continuous, differentiable or analytic functions
• 46H05: General theory of topological algebras
• 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 210
• Numer: 3
• Strony: 227-237

Daty

wydano

2012

Twórcy

autor

M. R. Koushesh

• Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran
• School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Identyfikatory

DOI

10.4064/sm210-3-3