We determine the first order asymptotics of the trace of $f(P_nUP_n)$ and the determinant ${\rm det}\,P_nUP_n$ for operators $U$ belonging to the F{\o}lner 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.