From Kleisli Categories to Commutative C*-Algebras: Probabilistic Gelfand Duality

Robert Furber, Bart Jacobs

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

C*-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C*-algebras. This paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras. This yields a new probabilistic version of Gelfand duality, involving the ``Radon'' monad on the category of compact Hausdorff spaces. We also show that a commutative C*-algebra is isomorphic to the space of convex continuous functionals from its state space to the complex numbers. This allows us to obtain an appropriately commuting state-and-effect triangle for commutative C*-algebras.
Original languageEnglish
Title of host publicationAlgebra and Coalgebra in Computer Science
EditorsReiko Heckel, Stefan Milius
Place of PublicationBerlin, Heidelberg
PublisherSpringer Berlin Heidelberg
Pages141-157
Number of pages17
ISBN (Electronic)978-3-642-40206-7
ISBN (Print)978-3-642-40206-7
DOIs
Publication statusPublished - 3 Sep 2013
Event5th International Conference on Algebra and Coalgebra in Computer Science - Warsaw, Poland
Duration: 3 Sep 20136 Sep 2013

Publication series

NameLecture Notes in Computer Science
PublisherSpringer, Berlin, Heidelberg
Volume8089
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference5th International Conference on Algebra and Coalgebra in Computer Science
Abbreviated titleCALCO 2013
Country/TerritoryPoland
CityWarsaw
Period3/09/136/09/13

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