Furber, Robert W. J. and Jacobs, Bart P. F. - From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

lmcs:1565 - Logical Methods in Computer Science, June 10, 2015, Volume 11, Issue 2
From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

Authors: Furber, Robert W. J. and Jacobs, Bart P. F.

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. At first, 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 then show that the state space functor from C*-algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C*-algebras.


Source : oai:arXiv.org:1303.1115
DOI : 10.2168/LMCS-11(2:5)2015
Volume: Volume 11, Issue 2
Published on: June 10, 2015
Submitted on: February 21, 2014
Keywords: Mathematics - Category Theory,Computer Science - Logic in Computer Science


Share

Consultation statistics

This page has been seen 212 times.
This article's PDF has been downloaded 128 times.