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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Ultraproducts, $\epsilon$-multipliers, and isomorphisms
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by Michael Cambern and Krzysztof Jarosz PDF
Proc. Amer. Math. Soc. 105 (1989), 929-937 Request permission

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 $||T||||{T^{ - 1}}||$ 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.
References
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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