An index theorem for Toeplitz operators with noncommutative symbol space
The authors consider Toeplitz operators with symbol in a \(C^*\)-algebra \(A\) carrying an action \(\alpha\) of \(\mathbb{R}\). They prove that when \(A\) has an \(\mathbb{R}\)-invariant trace \(\tau\), 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 \(\tau\) and the infinitesimal generator of the one-parameter automorphism group \(\alpha\). 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.
Toeplitz operators with symbol in a \(C^*\)-algebra
scientific article; zbMATH DE number 599765