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

convex characteristics of quaternionic positive definite functions on abelian groups