Duality for convex monoids

TL;DR

This paper investigates the duality between effect modules and convex spaces of states in finite-dimensional Hopf algebra contexts, revealing how convex monoids arise from group and function algebras and their combinations.

Contribution

It demonstrates how convex monoids associated with certain Hopf algebras can be derived from density matrices on irreducible representations, extending Kadison duality.

Findings

Convex monoids from function and group algebras are interconnected via coproducts.

The structure of convex monoids can be obtained from density matrices on irreducible representations.

The duality extends to tensor products of group and function algebras.

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.