Bounded and unitary elements in Pro-C*-algebras

• Autores: Rachid El Harti, Gábor Lukács
• Localización: Applied categorical structures, ISSN 0927-2852, Vol. 14, Nº 2, 2006, págs. 151-164
• Idioma: inglés
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• Resumen

  • 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-Cech-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.