On p-Convexity and q-Concavity in Non-Commutative Symmetric Spaces

Abstract

It is shown that if a symmetric Banach space E on the positive semi-axis is p-convex (q-concave) then so is the corresponding non-commutative symmetric space E(τ) of τ-measurable operators affiliated with some semifinite von Neumann algebra \({({\mathcal{M}}, \tau)}\) , with preservation of the convexity (concavity) constants in the case that \({{\mathcal{M}}}\) is non-atomic. Similar statements hold in the case that E satisfies an upper (lower) p-estimate and extend to the more general semifinite setting earlier results due to Arazy and Lin for unitary matrix spaces.