On unbounded p-summable Fredholm modules

摘要

We prove that odd unbounded p-summable Fredholm modules are also bounded p-summable Fredholm modules (this is the odd counterpart of a result of A. Connes for the case of even Fredholm modules). The approach we use is via estimates of the form parallel to phi(D) - phi(D-o)parallel to L-p(M,L-tau) less than or equal to C . parallel to D - D-o parallel to (1/2), where phi(t) = t(1 + t(2)) (-1/2), D-o = D-o* is an unbounded linear operator affiliated with a semifinite von Neumann algebra M, D - D-o is a bounded self-adjoint linear operator from M and (1 + D-o(2)) (-1/2) is an element of L-p(M,tau), where L-p(M,tau) is a non-commutative L-p-space associated with M. It follows from our results that if p is an element of (1, z), then phi(D) - phi(D-o) belongs to the space L-p(M, tau). (C) 2000 Academic Press. [References: 38]