The unbounded case is proved by reducing to the bounded case via the map We prove simultaneously a type II version of our results. We also prove a bounded finitely summable version of the form: for an integer. If and are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. For any piecewise smooth path with and unitarily equivalent we show that the integral of the 1-form. Then we show that for a sufficiently large half-integer: is a closed 1-form. Now, for in our manifold, any is given by an in as the derivative at along the curve in the manifold. Getzler in the -summable case) to consider the operator as a parameter in the Banach manifold,, so that spectral flow can be exhibited as the integral of a closed 1-form on this manifold. A subshift X is called minimal if it verifies one of the following equivalent conditions: There is no subshift Y such that (Ysubsetneq X). We construct a binary minimal subshift whose words of length n form a connected subset of the Hamming graph for each n. This integer, recovers the pairing of the -homology class with the -theory class. The spectral flow of this path (or ) is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as runs from 0 to 1. More precisely, we show that is a norm-continuous path of (bounded) self-adjoint Fredholm operators. We show that when is a subshift associated to a substitution, the group is an extension of by a finite group for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of. The path is a “continuous” path of unbounded self-adjoint “Fredholm” operators. We study, focusing on the case when is a minimal subshift. If is a unitary in the dense -subalgebra mentioned in (2) then where is a bounded self-adjoint operator. Abstract: An odd unbounded (respectively, -summable) Fredholm module for a unital Banach -algebra,, is a pair where is represented on the Hilbert space,, and is an unbounded self-adjoint operator on satisfying: (1) is compact (respectively, Trace, and (2) is a dense - subalgebra of.
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