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 ${\mathcal G} = M _0 \times M _0 \cup \mathbb{R}^q \times M _1 \times M _1$. Since it has been shown that for the case that $q \geqslant 3$ 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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