Isomorphisms of Finite Type II Rings of Operators

Abstract

A ring isomorphism between two rings of operators which preserves the * operation induces, by restriction, an isomorphism of the groups of unitary operators in the two rings. Some time ago, H. A. Dye raised the question whether the ring, * structure of a ring of operators was completely determined by the group structure of its unitary group. He proved in [2] that, for a certain class of rings of operators, the ring, * structure was determined by this group structure plus certain topological information about the group. Now, there is a simple way of associating unitary operators with projections; namely, send the projection P to the unitary I - 2P. This correspondence is a 1-1 map between all projections in a ring of operators and all unitaries of order 2 in the ring. Both Dye and I. Kaplansky were able to show that in the case of a factor of type II, this correspondence made it possible to reduce the original question about unitary groups to the question whether the ring, * structure was determined by the order and orthogonality relations in the set of projections in the ring. Previously to this, the present author had raised this very question, and, by applying J. von Neumann's techniques and results on regular rings and continuous geometries, had gotten the following theorem: if the lattices of projections of two semifinite rings of operators without 2-homogeneous summands are isomorphic via an isomorphism which preserves orthogonality, then the lattice isomorphism is induced by a ring, * isomorphism. Dye later proved the theorem independently, and was in fact able to dispense with the assumption of semifiniteness. Next came the problem of getting from here back to the theorem about unitary groups, in more general cases. Dye succeeded in so doing for factors (at least, those factors in which the theorem is true: those not of type 12n). In the present paper, we first prove the lattice of projections theorem for finite type II rings of operators (where the proof is relatively simple, given von Neumann's continuous geometry theorems), and then prove the unitary group theorem for this same class of rings of operators. 2. Preliminaries