The Heisenberg calculus, index theory and cyclic cohomology. (English) Zbl 1489.58008
In the present work, the authors present a local formula for the index of a Heisenberg elliptic operator on a compact contact manifold.
There have been several works on the index problem for Heisenberg elliptic operators, including the second author’s index theorem for Heisenberg elliptic operators in terms of \(K\)-theory [E. van Erp, Ann. Math. (2) 171, No. 3, 1647–1681 (2010; Zbl 1206.19004)]. However, it is a delicate problem to translate the \(K\)-theoretic index theorems for Heisenberg elliptic operators in [loc. cit.] into the local index formula, since the natural isomorphism between the \(K\)-theory groups adapted in [loc. cit.], which is obtained from noncommutative geometry, does not provide an explicit formula.
The works [C. Epstein and R. Melrose, “The Heisenberg algebra, index theory and homology”, Unpublished manuscript; Math. Res. Lett. 5, No. 3, 363–381 (1998; Zbl 0929.58012)] and [C. Epstein, “Lectures on indices and relative indices on contact and CR-manifolds”, Woods Hole Mathematics. 27–93 (2004)] by Epstein and Melrose enlightened the authors to an idea for solving the problem of finding the desired local index formula for general Heisenberg elliptic operators. Epstein and Melrose derived an index formula for a restricted class of operators, namely Hermite operators, but this result cannot be simply extended to more general Heisenberg elliptic operators due to some technical reasons. The authors resolve this issue by adapting the odd Chern character formula in the works of the first author [Commun. Math. Phys. 208, No. 1, 1–23 (1999; Zbl 0960.46047); Explicit formulae for characteristic classes in noncummutative geometry. The Ohio State University (PhD Thesis) (1999)], where the trace, connection and curvature in the formula are replaced by the newly introduced ones in the current work. The space of differential forms with values in the novel algebra constructed in Section 3 defines a recipient for the newly introduced connection and curvature, and the graded trace on this space of differential forms is constructed as well. Collecting all these notions enables authors to obtain the character map which is the main ingredient of the desired index formula. The construction of the character map strongly relies on the techniques from cyclic cohomology.
The proof of the main result, namely the local index formula for the Heisenberg elliptic operators, follows by demonstrating that in the case of Toeplitz operators the main result recovers Boutet de Monvel’s index theorem [L. Boutet de Monvel, Invent. Math. 50, 249–272 (1979; Zbl 0398.47018)] and by showing that every element in the domain of the character map mentioned above can be written in terms of Toeplitz operators. The appendix is devoted to the holomorphic closedness of the Fréchet algebra of order zero Heisenberg principal symbols, which induces an isomorphism between the \(K\)-theory groups of this Fréchet algebra and its \(C^*\)-algebraic closure.
There have been several works on the index problem for Heisenberg elliptic operators, including the second author’s index theorem for Heisenberg elliptic operators in terms of \(K\)-theory [E. van Erp, Ann. Math. (2) 171, No. 3, 1647–1681 (2010; Zbl 1206.19004)]. However, it is a delicate problem to translate the \(K\)-theoretic index theorems for Heisenberg elliptic operators in [loc. cit.] into the local index formula, since the natural isomorphism between the \(K\)-theory groups adapted in [loc. cit.], which is obtained from noncommutative geometry, does not provide an explicit formula.
The works [C. Epstein and R. Melrose, “The Heisenberg algebra, index theory and homology”, Unpublished manuscript; Math. Res. Lett. 5, No. 3, 363–381 (1998; Zbl 0929.58012)] and [C. Epstein, “Lectures on indices and relative indices on contact and CR-manifolds”, Woods Hole Mathematics. 27–93 (2004)] by Epstein and Melrose enlightened the authors to an idea for solving the problem of finding the desired local index formula for general Heisenberg elliptic operators. Epstein and Melrose derived an index formula for a restricted class of operators, namely Hermite operators, but this result cannot be simply extended to more general Heisenberg elliptic operators due to some technical reasons. The authors resolve this issue by adapting the odd Chern character formula in the works of the first author [Commun. Math. Phys. 208, No. 1, 1–23 (1999; Zbl 0960.46047); Explicit formulae for characteristic classes in noncummutative geometry. The Ohio State University (PhD Thesis) (1999)], where the trace, connection and curvature in the formula are replaced by the newly introduced ones in the current work. The space of differential forms with values in the novel algebra constructed in Section 3 defines a recipient for the newly introduced connection and curvature, and the graded trace on this space of differential forms is constructed as well. Collecting all these notions enables authors to obtain the character map which is the main ingredient of the desired index formula. The construction of the character map strongly relies on the techniques from cyclic cohomology.
The proof of the main result, namely the local index formula for the Heisenberg elliptic operators, follows by demonstrating that in the case of Toeplitz operators the main result recovers Boutet de Monvel’s index theorem [L. Boutet de Monvel, Invent. Math. 50, 249–272 (1979; Zbl 0398.47018)] and by showing that every element in the domain of the character map mentioned above can be written in terms of Toeplitz operators. The appendix is devoted to the holomorphic closedness of the Fréchet algebra of order zero Heisenberg principal symbols, which induces an isomorphism between the \(K\)-theory groups of this Fréchet algebra and its \(C^*\)-algebraic closure.
Reviewer: Gihyun Lee (Bonn)
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 32V40 | Real submanifolds in complex manifolds |
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