Abstract
Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that both these convex monoids can be obtained from the other one by taking a coproduct of density matrices on the irreducible representations. We will also show that the same holds for a tensor product of a group and a function algebra.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bennett, M.K., Foulis, D.J.: Effect algebras and unsharp quantum logics. Found. Phys. 24(10), 1331–1352 (1994). doi:10.1007/BF02283036. MR 1304942
Dvurečenskij, A., Pulmannová, S.: New trends in quantum structures, Mathematics and its Applications, vol. 516. Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava (2000). MR 1861369
Furber, R., Jacobs, B.: From Kleisli categories to commutative C*-algebras: probabilistic Gelfand duality. Logical methods in computer science 11(2:5), 1–28 (2015). doi:10.2168/LMCS-11(2:5)2015
Gudder, S.P., Pulmannová, S.: Representation theorem for convex effect algebras. Comment. Math. Univ. Carolin. 39(4), 645–659 (1998). MR 1715455
Jacobs, B., Mandemaker, J.: The Expectation Monad in Quantum Foundations, eprint. arXiv:1112.3805v2 (2012)
Mislove, M.: Probabilistic Monads, Domains and Classical Information, eprint. arXiv:1207.7150v1 (2012)
Takesaki, M.: Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125. Springer, Berlin (2003). Operator Algebras and Non-commutative Geometry, 6. MR 1943006
Timmermann, T.: An invitation to quantum groups and duality. European Mathematical Society (EMS), Zürich (2008). MR 2397671
Woronowicz, S.L.: Compact matrix pseudogroups. Comm. Math. Phys. 111 (4), 613–665 (1987). available at http://projecteuclid.org/euclid.cmp/1104159726. MR 901157
Semadeni, Z.: Categorical methods in convexity. In: Proceedings of the Colloquium on Convexity, pp. 281–307 (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Roumen, F., Roy, S. Duality for Convex Monoids. Order 34, 349–362 (2017). https://doi.org/10.1007/s11083-016-9404-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-016-9404-1