On $p$-convexification of the Banach-Kantorovich lattice

Abstract

Let $B$ be a complete Boolean algebra, $Q(B)$ the Stone compact of $B$, and let $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x: Q(B) \to [-\infty, +\infty]$, assuming possibly the values $\pm\infty$ on nowhere-dense subsets of $Q(B)$. Let $(E,\|\cdot\|_{E}) \subset C_\infty (Q(B))$ be a Banach-Kantorovich lattice over the algebra $L^0(\Omega)$ of equivalence classes of almost everywhere finite real-valued measurable functions on a measurable space $(\Omega, \Sigma, \mu)$ with $\sigma$-finite measure $\mu$. The paper defines the $p$-convexification of the Banach-Kantorovich lattice $(E,\|\cdot\|_{E})$ and proves that it is also a Banach-Kantorovich lattice over $L^0(\Omega)$.

Keywords

• p-convexification
• maharam measure
• banach-kantorovich space