144,008 research outputs found
Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary
We consider two calculi of pseudodifferential operators on manifolds with
fibered boundary: Mazzeo's edge calculus, which has as local model the
operators associated to products of closed manifolds with asymptotically
hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which
is similarly modeled on products of closed manifolds with asymptotically
Euclidean spaces. We construct an adiabatic calculus of operators interpolating
between them, and use this to compute the `smooth' K-theory groups of the edge
calculus, determine the existence of Fredholm quantizations of elliptic
symbols, and establish a families index theorem in K-theory
Pseudodifferential operators on manifolds with foliated boundaries
Let X be a smooth compact manifold with boundary. For smooth foliations on
the boundary of X admitting a `resolution' in terms of a fibration, we
construct a pseudodifferential calculus generalizing the fibred cusp calculus
of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a
simple description of the Fredholm operators inside the calculus. When the
leaves of the fibration `resolving' the foliation are compact, we also obtain
an index formula for Fredholm perturbations of Dirac-type operators. Along the
way, we obtain a formula for the adiabatic limit of the eta invariant for
invertible perturbations of Dirac-type operators, a result of independent
interest generalizing the well-known formula of Bismut and Cheeger.Comment: 49 pages, added references, strengthened the results, added an index
calculation for some quotients of gravitational instantons. To appear in the
Journal of Functional Analysi
Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras
We prove an index theorem for boundary value problems in Boutet de Monvel's
calculus on a compact manifold X with boundary. The basic tool is the tangent
semigroupoid \cT^-X generalizing the tangent groupoid defined by Connes in
the boundaryless case, and an associated continuous field C*_r(\cT^-X) of
C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with
the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are
isomorphic to the algebra K of compact operators. We therefore obtain a natural
map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show
that this is the analytic index map. On the other hand, using ideas from
noncommutative geometry, we construct the topological index map and prove that
it coincides with the analytic index map
A planar calculus for infinite index subfactors
We develop an analog of Jones' planar calculus for II_1-factor bimodules with
arbitrary left and right von Neumann dimension. We generalize to bimodules
Burns' results on rotations and extremality for infinite index subfactors.
These results are obtained without Jones' basic construction and the resulting
Jones projections.Comment: 56 pages, many figure
On the index of pseudo-differential operators on compact Lie groups
In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean-Singer index formula together with the operator-valued functional calculus developed here
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