Bounded and Unitary Elements in Pro-C*-algebras |
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Authors: | Rachid El Harti Gábor Lukács |
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Institution: | (1) Department of Mathematics, University Hassan I, FST de Settat, BP 577, 2600 Settat, Morocco;(2) Department of Mathematics and Statistics, Dalhousie University, Halifax, B3H 3J5, Nova Scotia, Canada |
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Abstract: | A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−)
b
that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−)
b
is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented. |
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Keywords: | pro-C*-algebra Gelfand duality Stone-Č ech-compactification Tychonoff space strongly functionally generated k-space k R -space bounded spectrally bounded coreflection exact |
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