The endomorphisms of the total operator domain of an infinite module

Abstract

The total operator domain of a finite vector space of dimensionality n over a division ring is a total matrix algebra of order n2 over a division ring anti-isomorphic with the given ring. The only nonzero endomorphisms of this total matrix algebra are the inner automorphisms, the automorphisms induced by automorphisms of the division ring, and products of these two(1). If we now start with an infinite vector space with Hamel basis over a division ring P, the total operator domain can be thought of as a matrix algebra of infinite order over a division algebra P anti-isomorphic with P. Since the choice of elements in the infinite matrices is restricted, the matrix algebra should perhaps not be called total, though it is a maximal ring contained in the set of all infinite matrices with elements in P. The present paper is a study of the endomorphisms of this total operator domain. To avoid the assumption of the well-ordering of any set, the infinite vector space is assumed to have a countable Hamel basis over P. Most of the methods introduced and the results carry over, however, for a basis of any cardinal number if the well-ordering assumption is used. As in the finite case, it is shown that the only nonzero endomorphisms are meromorphisms of the domain. However, the meromorphisms need not be automorphisms. A formula is given for all meromorphisms of the operator ring. Simplifications occur in the cases of P-meromorphisms and automorphisms. In the latter case the results are the same as those for the finite matrix algebras. The total operator domain has no nonzero anti-endomorphisms.