The first Szegö limit theorem for non-selfadjoint operators in the Følner algebra

Abstract

We determine the first order asymptotics of the trace of $f(P_nUP_n)$ and the determinant $\det P_nUP_n$ for operators $U$ belonging to the Følner algebra associated with the sequence $\{P_n\}$ and satisfying an "index zero" condition. We present three different proofs of the main result in the case where $U$ is a normal operator.