Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups

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 \({\mathbb {S}}\) $\mathbb{S}$ , i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when G is of exponent \(\le 2\) $\le 2$ . 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 results.