Banach envelopes in symmetric spaces of measurable operators (vol 21, pg 473, 2017)

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 are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let be a non-atomic, semifinite von Neumann algebra with a faithful, normal, -finite trace and E be as symmetric function space on or symmetric sequence space. We compute Banach envelope norms on and for any quasi-normed symmetric space E. Then we show under assumption that that the Banach envelope of is equal to isometrically. We also prove the analogous result for unitary matrix spaces C-E.