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We prove a
-equivariant version of the algebraic index theorem, where
is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.
Fix a von Neumann algebra
equipped with a suitable trace
. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with
. If the unbounded operator is
-summable (that is, its resolvents are contained in the ideal
), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of
-summable operators, and then using Laplace transforms to obtain a
-summable formula. In this paper, we present a direct proof of the
-summable formula that is both shorter and simpler than theirs.
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.
In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant
-theory with respect to a compact torus
of various spaces associated to a linear action of
in a vector space
can both be described using some vector spaces of distributions, on the dual of the group
or on the dual of its Lie algebra
. The morphism from
-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a
-transversally elliptic operator on
are determined using the infinitesimal index of the symbol.
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