On Unbounded p-Summable Fredholm Modules

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 ‖φ(D)−φ(D0)‖Lp(M,τ)⩽C·‖D−D0‖1/2, where φ(t)=t(1+t2)−1/2, D0=D*0 is an unbounded linear operator affiliated with a semifinite von Neumann algebra M, D−D0 is a bounded self-adjoint linear operator from M and (1+D20)−1/2∈Lp(M,τ), where Lp(M,τ) is a non-commutative Lp-space associated with M. It follows from our results that if p∈(1,∞), then φ(D)−φ(D0) belongs to the space Lp(M,τ).