An application of virtual degeneracy to two-valued subsets of $L_{p}$-spaces

Anthony Weston

Published 2014-12-29Version 1

Suppose $0 < p < 2$ and that $(\Omega, \mu)$ is a measure space for which $L_{p}(\Omega, \mu)$ is at least two-dimensional. Kelleher, Miller, Osborn and Weston have shown that if a subset $B$ of $L_{p}(\Omega, \mu)$ does not have strict $p$-negative type, then $B$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space). Examples show that the converse of this statement is not true in general. In this note we describe a class of subsets of $L_{p}(\Omega, \mu)$ for which the converse statement holds. We prove that if a two-valued set $B \subset L_{p}(\Omega, \mu)$ is affinely dependent (when $L_{p}(\Omega, \mu)$ is considered as a real vector space), then $B$ does not have strict $p$-negative type. This result is peculiar to two-valued subsets of $L_{p}(\Omega, \mu)$ and generalizes an elegant theorem of Murugan. It follows, moreover, that of certain types of isometry with range in $L_{p}(\Omega, \mu)$ cannot exist.