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Journal of Operator Theory

Volume 92, Issue 1,  Summer 2024  pp. 189-213.

Renormalized index formulas for elliptic differential operators on boundary groupoids

Authors:  Yu Qiao (1), Bing Kwan So (2)
Author institution:(1) School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, Shaanxi, China
(2) School of Mathematics, Jilin University, Changchun, 130023, Jilin, China


Summary:  We consider the index problem of certain boundary groupoids of the form G=M0×M0Rq×M1×M1. Since it has been shown that for the case that q is odd, K _0 (C^* ({\mathcal G})) \cong \mathbb Z , and moreover the K-theoretic index coincides with the Fredholm index, we attempt in this paper to derive a numerical formula for elliptic differential operators on renormalized groupoid. Our approach is similar to that of renormalized trace of Moroianu and Nistor. However, we find that when q \geqslant 3, the \eta term vanishes, and hence the K-theoretic and Fredholm indices 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.

DOI: http://dx.doi.org/10.7900/jot.2022sep12.2399
Keywords:  index formulas, Fredholm index, pseudodifferential operators, boundary groupoids

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