Applied Categorical Structures

, Volume 14, Issue 2, pp 151–164

Bounded and Unitary Elements in Pro-C*-algebras

Article

DOI: 10.1007/s10485-006-9012-0

Cite this article as:
El Harti, R. & Lukács, G. Appl Categor Struct (2006) 14: 151. doi:10.1007/s10485-006-9012-0

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.

Key words

pro-C*-algebraGelfand dualityStone-Čech-compactificationTychonoff spacestrongly functionally generatedk-spacekR-spaceboundedspectrally boundedcoreflectionexact

Mathematics Subject Classification (2000)

18A0546H0546J0546K05

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity Hassan I, FST de SettatSettatMorocco
  2. 2.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada