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 \({\mathbb {S}}\), 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 \(\le 2\). 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.

本文关注任意交换群G上的归一化四元值正定函数的拓扑空间，特别是其凸特征。有两个主要结果。首先，我们证明该函数族中的极值元素正是G到球群\({\mathbb {S}}\)，即四元数代数中的单位3-球面的同态。其次，我们揭示了一个新现象：除非G是指数\(\le 2\)，否则此类函数的紧凸集不是鲍尔单纯形。相反，众所周知，其复数对应物始终是鲍尔单纯形。我们还提供了应用程序等功能的完整表示和其他一些次要结果。