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When does $ {\rm ca}(\Sigma,X)$ contain a copy of $ l\sb \infty$ or $ c\sb 0$?


Author: Lech Drewnowski
Journal: Proc. Amer. Math. Soc. 109 (1990), 747-752
MSC: Primary 46E27; Secondary 46B20, 46G10
DOI: https://doi.org/10.1090/S0002-9939-1990-1012927-4
MathSciNet review: 1012927
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Abstract: For a $ \sigma $-algebra $ \Sigma $ and a Banach space $ X,ca\left( {\Sigma ,X} \right)$ is the Banach space of all vector measures from $ \Sigma $ to $ X$. If $ \Sigma $ admits a nonzero atomless finite positive measure, then $ ca\left( {\Sigma ,X} \right) \supset {l_\infty }$ (or $ {c_0}$) if and only if there is a noncompact bounded linear operator from $ {l_2}$ to $ X$ (Theorem 1). Otherwise, $ ca\left( {\Sigma ,X} \right) \supset {l_\infty }$ (or $ {c_0}$) if and only if $ X \supset {l_\infty }$ (or $ {c_0}$) (Theorem 2).


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DOI: https://doi.org/10.1090/S0002-9939-1990-1012927-4
Keywords: Banach space, vector measure, copy of $ {l_\infty }$, copy of $ {c_0}$, Hilbert space $ {l_2}$, noncompact bounded linear operator
Article copyright: © Copyright 1990 American Mathematical Society

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