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Representation and index theory for Toeplitz operators

We study Toeplitz operators on the Hardy spaces of connected compact abelian groups and of tube-type bounded symmetric domains. A representation theorem for these operators and for classes of abstract Toeplitz elements in C * -algebras is proved. This is used to give a unified treatment to index theory in this setting, and a variety of new index theorems are proved that generalize the Gohberg-Krein theorem for Toeplitz operators on the Hardy space of the unit circle in the plane.