Banach envelopes in symmetric spaces of measurable operators
Document Type
Article
Publication Date
3-1-2017
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 E^ 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 CE for any quasi-normed symmetric space E. Then we show under assumption that E∈ (HC) that the Banach envelope E(M, τ) ^ of E(M, τ) is equal to E^ (M, τ) isometrically. We also prove the analogous result for unitary matrix spaces CE.
Publication Title
Positivity
Volume
21
Issue
1
First Page
473
Last Page
492
Digital Object Identifier (DOI)
10.1007/s11117-016-0430-4
ISSN
13851292
E-ISSN
15729281
Citation Information
CzerwiAska, & KamiAska, A. (2016). Banach envelopes in symmetric spaces of measurable operators. Positivity : an International Journal Devoted to the Theory and Applications of Positivity in Analysis, 21(1), 473–492. https://doi.org/10.1007/s11117-016-0430-4