2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99781We establish an index theorem for Toeplitz operators on odd dimensional spin manifolds with boundary. It may be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of $η$ type associated to $K^1$ representatives on even dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.Completely revised. A gap in the proof fixed. To appear in JFADifferential Geometry58Gxxtext