Integral Formula for Spectral Flow for -Summable Operators

Abstract

Abstract. Fix a von Neumann algebra N equipped with a suitable trace τ. For a path of self-adjoint Breuer-Fredholm operators, the spectral ow measures the net amount of spectrum which moves from negative to non-negative. We consider speci cally the case of paths of bounded perturbations of a xed unbounded self-adjoint BreuerFredholm operator a liated with N. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal L p ), then it is possible to obtain an integral formula which calculates spectral ow. is integral formula was rst proven by Carey and Phillips, building on earlier approaches of Phillips.eir proof was based on rst obtaining a formula for the larger class of θ-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.