Unbounded Fredholm modules and spectral flow

An odd unbounded (respectively, p-summable) Fredholm module for a unital Banach *-algebra, A, is a pair (H, D) where A is represented on the Hilbert space, H, and D is an unbounded self-adjoint operator on H satisfying: (1) (1 + D-2)(-1) is compact (respectively, Trace ((1 + D-2)(-(p/2))) < infinity), and (2) {a is an element of A \ [D, a] is bounded} is a dense *-subalgebra of A. If u is a unitary in the dense *-subalgebra mentioned in (2) then uDu* = D + u[D, u*] = D + B where B is a bounded self-adjoint operator. The path D-t(u) := (1 - t)D + tuDu* = D + tB is a "continuous" path of unbounded self-adjoint "Fredholm" operators. More precisely, we show that F-t(u) := D-t(u) (1 + (D-t(u))(2))(-1/2) is a norm-continuous path of(bounded) self-adjoint Fredholm operators. The spectral flow of this path {F-t(u)} (or {D-t(u)}) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as t runs from 0 to 1. This integer, sf({D-t(u)}):= sf({F-t(u)}), recovers the pairing of the K-homology class [D] with the K-theory class [u]. We use I. M. Singer's idea (as did E. Getzler in the theta-summable case) to consider the operator B as a parameter in the Banach manifold, B-sa(H), so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. Now, for B in our manifold, any X is an element of T-B (B-sa(H)) is given by an X in B-sa(H) as the derivative at B along the curve t bar right arrow B + tX in the manifold. Then we show that for m a sufficiently large half-integer: alpha(X) = 1/(C) over tilde(m) Tr(X(1 +(D + B)(2))(-m)) is a closed 1-form. For any piecewise smooth path {D-t = D + B-t} with D-0 and D-1 unitarily equivalent we show that sf({D-1}) = 1/(C) over tilde(m) integral(0)(1) Tr(d/dt(D-t)(1 + D-t(2))(-m))dt the integral of the 1-form alpha. If D-0 and D-1 are not unitarily equivalent, we must add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form: sf({F-t}) = 1/C-n integral(0)(1) Tr(d/dt(F-t)(1 - F-t(2))(n)) dt for n greater than or equal to p-1/2 integer. The unbounded case is proved by reducing to the bounded case via the map D bar right arrow F = D(1 + D-2)(-1/2). We prove simultaneously a type II version of our results.