convex characteristics of quaternionic positive definite functions on abelian groups

TL;DR

This paper investigates the convex structure of quaternionic positive definite functions on abelian groups, revealing unique properties of their extreme points and convex sets that differ from the complex case.

Contribution

It characterizes the extreme elements as homomorphisms to the quaternion sphere and uncovers a non-Bauer simplex phenomenon in the quaternionic setting.

Findings

Extreme elements are homomorphisms to the quaternion sphere

The convex set is not a Bauer simplex unless G has exponent ≤ 2

Provides an integral representation for these functions

Abstract

This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group S, i.e., the unit 3-sphere in the quaternion algebra. Secondly, we real a phenomenon which does not exist in the complex setting: The compact convex set of such functions is not a Bauer simplex except when G is of exponent less than or equal to 2. In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor interesting results.