Banach Envelopes in Symmetric Spaces of Measurable Operators

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 $C_E$.