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Duality for convex monoids

Frank Roumen, Sutanu Roy

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.

Duality for convex monoids

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.

Paper Structure

This paper contains 7 sections, 14 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Effect algebras and modules
  4. Convex spaces
  5. Kadison duality for group and function algebras
  6. Convex Pontryagin duality for group and function algebras
  7. Convex Pontryagin duality for a tensor product

Key Result

Theorem \oldthetheorem

The compositions $C \circ \mathop{\mathrm{Spec}}\nolimits$ and $\mathop{\mathrm{Spec}}\nolimits \circ C$ are naturally equivalent to the identity functor. Hence the categories $\mathbf{KHaus}\xspace$ and $\mathbf{cC^*}\xspace$ are dually equivalent.

Theorems & Definitions (30)

  • Theorem \oldthetheorem: Gelfand
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 20 more