A note on traces of operators on $L^{\infty}(\lambda)$

Let (\Omega,\Sigma,\lambda) be a finite measure space and \tau(L^{\infty}(\lambda),L^{1}(\lambda)) be a natural Mackey topology on L^{\infty}(\lambda) . Let T:L^{\infty}(\lambda)\rightarrow L^{\infty}(\lambda) be a Bochner representable operator, that is, there exists g\in L^{1}(\lambda,L^{\infty}(\lambda)) so that T(u)=\int_{\Omega} u(\omega)g(\omega)\,d\lambda(\omega) for u\in L^{\infty}(\lambda) . It is shown that T is a nuclear operator between the locally convex space (L^{\infty}(\lambda),\tau(L^{\infty}(\lambda),L^{1}(\lambda))) and the Banach space L^{\infty}(\lambda) and T has a well-defined trace: \operatorname{tr}T=\int_{\Omega} g(\omega)(\omega)\,d\lambda(\omega) .