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

Abstract

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 $T:L^{\infty }(\lambda )\to L^{\infty }(\lambda )$ 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: $\mathrm{tr}T={\int }_{\Omega }g(\omega )(\omega )d\lambda (\omega )$.