An Index Theorem for Toeplitz Operators with Noncommutative Symbol Space

We consider Toeplitz operators with symbol in a C*-algebra A carrying an action α of R. We prove that when A has an R-invariant trace τ, the Toeplitz operators with invertible symbols are Fredholm elements of an appropriate von Neumann algebra, and give a formula for their Breuer index in terms of τ and the infinitesimal generator of the one-parameter automorphism group α. This theorem includes as special cases recent results of Curto-Muhly-Xia, Ji, and Lesch, as well as classical results of Gohberg-Krein and Coburn-Douglas-Schaeffer-Singer, and provides a more direct approach to the corresponding index theorems for systems.