Coherence invariant mappings of symmetric transformations

Abstract

Introduction. In a previous paper [3] a characterization was found of the coherence invariant mappings of the group of linear transformations of finite rank of one vector space onto a similar group for a second vector space. In this paper we shall investigate the coherence invariant mappings of the set of symmetric (self-adjoint) transformations of finite rank on a single vector space. We assume here that X is a vector space of dimension >3 over a field of characteristic '2 and that X is self-dual relative to a nondegenerate hermitian scalar product. The principal result obtained in this setting is that, similar to the above mentioned case, the coherence invariant mapping is essentially induced by a semilinear transformaton on X. L. K. Hua has proved this for the special case of symmetric matrices over a field, i.e., the case in which the scalar product is symmetric [i]. The generalization, therefore, is to the entire class of hermitian products and to infinite dimension and is made possible by an application of the fundamental theorem of projective geometry and techniques requiring commutativity of the field. The methods used differ from those employed by Hua and considerably shorten some of the arguments. The author wishes to acknowledge his indebtedness to Professor C. E. Rickart for the many suggestions he offered during the preparation of this paper.