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- Title
Banach envelopes in symmetric spaces of measurable operators.
- Authors
Czerwińska, M.; Kamińska, A.
- 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 $$\widehat{E}$$ 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 $$[0,\tau (\mathbf 1 ))$$ 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 $$\widehat{E(\mathcal {M},\tau )}$$ of $$E(\mathcal {M},\tau )$$ is equal to $$\widehat{E}\left( \mathcal {M},\tau \right) $$ isometrically. We also prove the analogous result for unitary matrix spaces $$C_E$$ .
- Subjects
SYMMETRIC spaces; BANACH algebras; VON Neumann algebras; MATRICES (Mathematics); NONCOMMUTATIVE function spaces
- Publication
Positivity, 2017, Vol 21, Issue 1, p473
- ISSN
1385-1292
- Publication type
Article
- DOI
10.1007/s11117-016-0430-4