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1. CJM Online first
Spectral flow for nonunital spectral triples We prove two results about nonunital index theory left open in a
previous paper. The
first is that the spectral triple arising from an action of the reals on a $C^*$-algebra with invariant trace
satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths
arising from the odd
index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.
Keywords:spectral triple, spectral flow, local index theorem Category:46H30 |
2. CJM 2012 (vol 65 pp. 331)
Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces We show that for spaces with 1-unconditional bases
lushness, the alternative Daugavet property and numerical
index 1 are equivalent. In the class of rearrangement
invariant (r.i.) sequence spaces the only examples of spaces with
these properties are $c_0$, $\ell_1$ and $\ell_\infty$.
The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$;
the same space is the only r.i. separable function space on $[0,1]$
with the Daugavet property over the reals.
Keywords:lush space, numerical index, Daugavet property, KÃ¶the space, rearrangement invariant space Categories:46B04, 46E30 |
3. CJM 2000 (vol 52 pp. 695)
Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
Ray-Singer-Quillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.
Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant lines Categories:58J52, 58J35, 58J20 |