Nuclear operators and Baire vector measures

Abstract

Let $C_b(X)$ be the Banach lattice of all bounded continuous real functions on a completely regular Hausdorff space $X$ and let $(E,\|\cdot\|_E)$ be a real Banach space. A linear operator $T:C_b(X)\to E$ is said to be $\sigma$-additive if $\|T(u_n)\|_E\to 0$ whenever $(u_n)$ is a sequence in $C_b(X)$ such that $u_n(x)\downarrow 0$ for each $x\in X$. We characterize $\sigma$-additive nuclear operators $T:C_b(X)\to E$ in terms of their representing Baire vector measures. The formula for the trace of $\sigma$-additive nuclear operators $T:C_b(X)\to C_b(X)$ is derived.