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 \(C_E\).
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We are grateful to the reviewer for his/hers insightful report of the paper, which allowed us to improve significantly the final version of the manuscript.
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An erratum to this article is available at http://dx.doi.org/10.1007/s11117-016-0435-z.
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Czerwińska, M.M., Kamińska, A. Banach envelopes in symmetric spaces of measurable operators. Positivity 21, 473–492 (2017). https://doi.org/10.1007/s11117-016-0430-4
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DOI: https://doi.org/10.1007/s11117-016-0430-4
Keywords
- Symmetric spaces of measurable operators
- Noncommutative function spaces
- Unitary matrix spaces
- Banach envelopes
- Mackey completion