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. This integer, recovers the pairing of the -homology class with the -theory class. MINIMAL SUBSHIFT OF FULL MEAN DIMENSION 5 We note that the mean dimension of ((0, D ) Z, ) is equal to D, where D is a positiveinteger or +. Moreover, enlarging the alphabet and using the characterization of the minimal subshifts in terms of homogeneous sequences, we. 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. Assume that T 2Homeo(C) is a minimal subshift, i.e., T acts as a shift on the Cantor set AZ for some nite alphabet Aand there exists a clopen T-invariant subset X AZ such that the action of T on X is minimal. More precisely, we show that is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The path is a “continuous” path of unbounded self-adjoint “Fredholm” operators. 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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