The longitudinal cocycle and the index of Toeplitz operators

AbstractLet D be a self-adjoint leafwise elliptic operator on a foliated manifold. Compressing multiplication operators to the range of the positive spectral projection for D yields the class of leafwise Toeplitz operators. The extension generated by these operators is constructed. A topological formula for the index of a Toeplitz operator with invertible symbol is given. This index can also be obtained by pairing the K-theory class of the symbol with a certain cyclic cocycle. If one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle, then this cocycle can be used to obtain refined invariants of the original operator