Title

Banach envelopes in symmetric spaces of measurable operators

Document Type

Article

Publication Date

3-1-2017

Subject Area

ARRAY(0x5577facca0d0)

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

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