Operator Algebras Related to the Bochner-Martinelli Integral
by
Nikolai Tarkhanov
Inst. of Math., Univ. of Potsdam, P.O. Box 60 15 53, 14415 Potsdam

The theory of one-dimensional singular integral equations with Cauchy kernel in the complex plane has essentially enriched the general Fredholm theory. It actually initiated the study of pseudodifferential operators on manifolds with singularities. However, singularities in higher dimensions are much more tricky than those in the complex plane. This motivates the study of concrete C^* -algebras generated by classical singular integral operators on higher-dimensional surfaces, which could give a powerful source of intuition to forecast the behaviour of general pseudodifferential operators. As but one example we show the C^* -algebra generated by the singular Bochner-Martinelli integral. In [1] the Clifford analysis is applied to derive an explicit formula for the square of the singular Bochner-Martinelli integral M over a smooth hypersurface S in Cⁿ. It reads M² = 1/4 + å_j [`a]<sub>j</sub> a<sub>j</sub>, where a<sub>j</sub> are certain singular integral operators on S and the index j runs from 1 to n(n-1)/2. Had we a solution X of the operator equations X<sup>2</sup> = - å<sub>j</sub> [`a]_j a_j and M X + X M = 0, the operators P^± = 1/2 ±M + X would be two independent projections on L² (S), whose sum is 1 + 2 X and the difference 2 M. Since M² + X² = 1/4 we would not change drastically the C^* -algebra generated by M, by adding the operator X » Ö{1/4 - M²} to the generators. On the other hand, the structure of the C^* -algebra with identity generated by two orthogonal projections is well understood, cf. Halmos (1969), etc. The aim of this paper is to bring together two areas in which elliptic theory plays an important role. The first area is the multidimensional complex analysis studying qualitative properties of solutions to the overdetermined elliptic Cauchy-Riemann system. The second one is the geometric analysis which deals with Dirac operators, i.e., first order matrix factorisations of the Laplace operator. Our approach invokes the diversity of Clifford algebra structures in Cⁿ to find an adequate representation of M². In this way we produce some concrete realisations of the algebra generated by the singular Bochner-Martinelli integral.

References

[1] R. Rosha-Chávez, M. Shapiro, and F. Sommen, On the singular Bochner-Martinelli integral, Integr. Equ. Oper. Theory 32 (1998), 354-365.

Date received: March 29, 2005