Deformation quantization and boundary value problems (Q2795355)

This paper introduces a natural star product on a compact symplectic manifold with boundary. Let us recall that deformation quantization manifolds with boundary has first been considered by \textit{R. Nest} and \textit{B. Tsygan} [J. Reine Angew. Math. 481, 27--54 (1996; Zbl 0866.58038)]. Their construction does not contain any boundary components, and there is no trace on the algebra which vanishes on commutators. In this work, the author concentrates on stronger geometric assumptions for the symplectic manifold. He constructs an algebra with trace which is the unity element of the algebra. It leads to an index formula. The index theorem is formulated as a conjecture. The classical harmonic oscillator is investigated explicitly.