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Ultraproducts, $ \epsilon$-multipliers, and isomorphisms


Authors: Michael Cambern and Krzysztof Jarosz
Journal: Proc. Amer. Math. Soc. 105 (1989), 929-937
MSC: Primary 46E40; Secondary 46B20, 46M99
DOI: https://doi.org/10.1090/S0002-9939-1989-0965240-7
MathSciNet review: 965240
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Abstract: For a compact Hausdorff space $ X$ and Banach dual $ {E^ * }$, denote by $ C(X,({E^ * },{\sigma ^ * }))$ the Banach space of all continuous functions on $ X$ to $ {E^ * }$ when the latter space is provided with its weak* topology. We show that if $ E_i^ * , i = 1,2$, belong to a class of Banach duals satisfying a condition involving the space of multipliers on $ E_i^ * $, then the existence of an isomorphism $ T$ mapping $ C({X_1},(E_1^ * ,{\sigma ^ * }))$ onto $ C({X_2},(E_2^ * ,{\sigma ^ * }))$ with $ \vert\vert T\vert\vert\vert\vert{T^{ - 1}}\vert\vert$ small implies that $ {X_1}$ and $ {X_2}$ are homeomorphic. Ultraproducts of Banach spaces and the notion of $ \varepsilon $-multipliers play key roles in obtaining this result.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0965240-7
Article copyright: © Copyright 1989 American Mathematical Society

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