RENORMALIZED INDEX FORMULAS FOR ELLIPTIC DIFFERENTIAL OPERATORS ON BOUNDARY GROUPOIDS

We consider the index problem of certain boundary groupoids of the form G = M-0 x M-0 boolean OR R-q x M-1 x M-1. Since it has been shown that for the case that q >= 3 is odd, K-0 (C* (G) congruent to 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 G. Our approach is similar to that of renormalized trace of Moroianu and Nistor. However, we find that when q >= 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.