Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRenormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider the index problem of certain boundary groupoids of the form $cG = M _0 times M _0 cup mathbb{R}^q times M _1 times M _1$. Since it has been shown that when $q $ is odd and $geq 3$, $K _0 (C^* (cG)) cong bbZ $, and moreover the $K$-theoretic index coincides with the Fredholm index, in this paper we attempt to derive a numerical formula. Our approach is similar to that of renormalized trace of Moroianu and Nistor cite{Nistor;Hom2}. However, we find that when $q geq 3$, the eta term vanishes, and hence the $K$-theoretic and Fredholm indexes of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the $q=1$ case we find that the result depends on how the singularity set $M_1$ lies in $M$.
rate research
Read More
This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators.
We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules.
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined subsymbol of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.
Let $A$ be a graded C*-algebra. We characterize Kasparovs K-theory group $hat{K}_0(A)$ in terms of graded *-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.
Noncommutative Hermitian structures were recently introduced by the second author as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a compact quantum homogeneous Hermitian space, which gives a natural set of compatibility conditions between covariant Hermitian structures and Woronowiczs theory of compact quantum groups. Each such object admits a Hilbert space completion, which possesses a remarkably rich yet tractable structure. The spectral behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operators by a negative (anti-ample) line bundle is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. When this is so, the operators index coincides with the holomorphic Euler characteristic of the underlying noncommutative complex structure. Our motivating family of examples, the irreducible quantum flag manifolds endowed with their Heckenberger-Kolb calculi, are presented in detail. The noncommutative Bott-Borel-Weil theorem is used to produce a family of Dolbeault-Dirac Fredholm operators for each quantum flag. Moreover, following previous spectral calculations of the authors, the Dolbeault-Dirac operator of quantum projective space is exhibited as a spectral triple in the sense of Connes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
