This is an announcement for the paper “Banach Envelopes in Symmetric Spaces of Measurable Operators” by Malgorzata Czerwinskahttp://arxiv.org/find/math/1/au:+Czerwinska_M/0/1/0/all/0/1, Annna Kaminskahttp://arxiv.org/find/math/1/au:+Kaminska_A/0/1/0/all/0/1.
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 $\hat {E}$ 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 $\hat{E(\mathcal{M}, \tau)}$ of $E(\mathcal{M}, \tau)$ is equal to $\hat{E(\mathcal{M}, \tau)}$ isometrically. We also prove the analogous result for unitary matrix spaces $C_E$.
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1606.00319