THE INDEX OF A PAIR OF PROJECTIONS

We discuss pairs of self-adjoint projections with P - Q is-an-element-of J(2n+1), the trace ideal, and prove that for m greater-than-or-equal-to n, tr(P - Q)2m+1 = tr(P - Q)2n+1 = dim(Ker Q and Ran P) - dim(Ker P and Ran Q) is an integer. We also prove that there exists a unitary V interchanging P and Q if and only if this integer is 0. (C) 1994 Academic Press, Inc.