Traces of Commutators of Integral Operators

This paper concerns a rather concrete phenomenon in abstract operator algebras. The main examples of the algebras we study are algebras of singular integral operators (pseudo-differential operators of order zero). As everyone knows, the Fredhohn index of a pseudo-differential operator depends only on its symbol and the Atiyah-Singer Index Theorem gives an explicit formula for computing the dependence. What we are doing might be though of analogously. The observation behind this paper is that traces of commutators or appropriate higher commutators depend only on "symbols"; then we compute the dependence in one and two dimensions. I t emerges that these considerations are closely related to index theory. The simplest type of operator system which we study is an almost commuting (a.c.) pair of operators on Hilbert space H; that is, a pair X, Y of bounded selfadjoint operators on H with trace class commutator [X, Y] = X F FX. The two main examples of almost commuting pairs are Toeplitz (or Wiener-Hopf) operators with smooth symbol and singular integral operators on the line. (In fact, the singular integral operators and multiplications provide generic examples [14], [17], [19].) In [8] the authors considered an algebra ~ of operators generated by an almost commuting pair and gave a quite satisfying formula for the trace of any commutator [A, B] from ~ in terms of the "symbol" of .4