1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.

While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.

In this note we are concerned with several questions on positive definite functions over a Hausdorff locally compact group. The main result, Theorem A, gives some necessary and sufficient conditions for to be a positive definite function when μ is a (complex Radon) measure. In particular, is a positive definite function if and only if μ ∊ L2, and Theorem B then follows by giving a complete characterization of functions of the type , where f ∊ L2. Perhaps the most interesting aspect of these results is that they provide further examples of results over a non-abelian, non-compact group, which otherwise are simple consequences (with μ, a bounded measure in Theorem A) of the theorems of Plancherel and Bochner.

Unless otherwise specified, all notation and definitions will follow [1;2]. The underlying group will always be G, a Hausdorff locally compact group with identity e, and with left Haar measure dx.