Lattice Homomorphisms on $L^{p}$-Spaces

Abstract

Let $(\Omega,\Sigma,\mu)$ be a complete $\sigma$-finite measure space and let $1\leq p\leq\infty$. We study bounded lattice homomorphisms $T:L^{p}(\mu)\to F$ into a Banach lattice $F$. To avoid the obstruction $1\notin L^{1}(\mu)$ when $\mu(\Omega)=\infty$, we work on the $\delta$-ring $\Sigma_{f}=\{A\in\Sigma:\mu(A)<\infty\}$ and the dense sublattice $S_{f}$ of simple functions with finite-measure support. For $1\leq p<\infty$ we show that every bounded lattice homomorphism induces a local Boolean set function $\nu:\Sigma_{f}\to F_{+}$ via $\nu(A)=T(\chi_{A})$, and that $T$ is the unique bounded extension of the associated simple-function integral $I_{\nu}$ from $S_{f}$ to $L^{p}(\mu)$. We introduce a $p$-variation functional $\|\nu\|_{(p)}$ and prove the intrinsic norm identity $\|T\|=\|\nu\|_{(p)}$ together with a converse construction theorem. When $F$ is an AL-space, boundedness is characterised by a Radon--Nikod\'ym derivative $g\in L^{q}(\mu)$ and $\|T\|=\|g\|_{q}$. For $p=1$ on finite measure spaces we also present an order-integral (Kantorovich--Wright type) formulation and show it agrees with the norm-density approach. Finally, we treat the case $p=\infty$ under $\sigma$-order continuity.