Banach Envelopes in Symmetric Spaces of Measurable Operators

Abstract

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class $(HC)$ of quasi-normed symmetric sequence or function spaces $E$ for which their Banach envelopes $\widehat{E}$ are also symmetric spaces. The class of symmetric spaces satisfying $(HC)$ contains but is not limited to order continuous spaces. Let $\mathcal{M}$ be a non-atomic, semifinite von Neumann algebra with a faithful, normal, $\sigma$-finite trace $\tau$ and $E$ be as symmetric function space on $[0,\tau(1))$ or symmetric sequence space. We compute Banach envelope norms on $E(\mathcal{M},\tau)$ and $C_E$ for any quasi-normed symmetric space $E$. Then we show under assumption that $E\in (HC)$ that the Banach envelope $\widehat{E(\mathcal{M},\tau)}$ of $E(\mathcal{M},\tau)$ is equal to $\widehat{E}(\mathcal{M},\tau)$ isometrically. We also prove the analogous result for unitary matrix spaces $C_E$.