Banach Envelopes in Symmetric Spaces of Measurable Operators

TL;DR

This paper characterizes when Banach envelopes of symmetric spaces of measurable operators remain symmetric, computes their norms, and establishes isometric relations between envelopes of classical and noncommutative spaces.

Contribution

It introduces the class (HC) of symmetric spaces whose Banach envelopes are also symmetric, and proves isometric relations for envelopes in noncommutative settings.

Findings

Characterization of class (HC) of symmetric spaces.

Computation of Banach envelope norms for noncommutative spaces.

Isometric equivalence of Banach envelopes in classical and noncommutative spaces.

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 $C_E$.