Banach Envelopes in Symmetric Spaces of Measurable Operators

2016/06/01 by Czerwinska, Malgorzata, Kaminska, Annna · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1606.00319

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 \widehatE are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let M be a non-atomic, semifinite von Neumann algebra with a faithful, normal, σ-finite trace τ and E be as symmetric function space on [0,τ(1)) or symmetric sequence space. We compute Banach envelope norms on E(M,τ) and C_E for any quasi-normed symmetric space E. Then we show under assumption that E∈ (HC) that the Banach envelope \widehatE(M,τ) of E(M,τ) is equal to \widehatE(M,τ) isometrically. We also prove the analogous result for unitary matrix spaces C_E.

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