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Register a new userThe principal-symbol index map for an algebra of pseudodifferential operators
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Authors
Severino T. Melo
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A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map is given. That index map takes values in K_0 of the commutator ideal E of the algebra, which is isomorphic to Z^2. It maps the K_1-class of an operator invertible modulo E to the Fredholm indices of a pair of elliptic pseudodifferentail operators on SxB, where S denotes the circle.
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Let $Gamma$ be a compact group acting on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. When $Gamma$ is finite, we explicitly characterize the operators $P$ for which the map $pi_alpha(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Gamma$ on the vector bundles $E_i$. When $Gamma = {1}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
Suppose that $phi$ and $psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_phi$ and $T_psi$ denote the Toeplitz operators with symbols $phi$ and $psi$ respectively. We give an explicit formula for the determinant of $T_phi T_psi T_phi^{-1} T_psi^{-1}$ in terms of the products of the tame symbols of $phi$ and $psi$ on the open unit disk.
Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,xi) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A is commutative. In this paper, we show that, if O is surjective for A, then it is also surjective for the algebra of k-by-k matrices with entries in A.
In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation $C^*$-algebra. To this end, we give the construction of a parametrix.
We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two non-compact, but topologically simpler manifolds. As another application we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.
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