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Unbounded Fredholm Modules and Spectral Flow
Published online by Cambridge University Press: 20 November 2018
Abstract
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 $\left( {{\left( 1+{{D}^{2}} \right)}^{-(p/2)}} \right)<\infty $, and
(2) $\{a\in A|\,\,[D,a]\,\,\text{is}\,\,\text{bounded}\}\text{ }$ is a dense $*$- subalgebra of $A$.
If $u$ is a unitary in the dense $*$-subalgebra mentioned in (2) then
where $B$ is a bounded self-adjoint operator. The path
is a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show that
is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path $\left\{ F_{t}^{u} \right\}$ (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,
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, ${{B}_{\text{sa}}}\left( H \right)$, 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 $X\,\in \,{{T}_{B}}({{B}_{\text{sa}}}(H))$ is given by an $X$ in ${{B}_{\text{sa}}}(H)$ as the derivative at $B$ along the curve $t\,\mapsto \,B\,+\,tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer:
is a closed 1-form. For any piecewise smooth path $\left\{ {{D}_{t}}=D+{{B}_{t}} \right\}$ with ${{D}_{0}}$ and ${{D}_{1}}$unitarily equivalent we show that
the integral of the 1-form $\alpha $. If ${{D}_{0}}$ and ${{D}_{1}}$ are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:
for $n\ge \frac{p-1}{2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D{{(1+{{D}^{2}})}^{-\frac{1}{2}}}$ We prove simultaneously a type II version of our results.
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- Copyright © Canadian Mathematical Society 1998
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