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Extreme operators in the unit ball of $ L(C(X),C(Y))$ over the complex field


Author: Alan Gendler
Journal: Proc. Amer. Math. Soc. 57 (1976), 85-88
MSC: Primary 47D20
DOI: https://doi.org/10.1090/S0002-9939-1976-0405173-9
MathSciNet review: 0405173
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Abstract: Assume that $ X$ and $ Y$ are compact Hausdorff spaces and that $ C(X)$ and $ C(Y)$ are the Banach spaces of continuous complex-valued functions on $ X$ and $ Y$, respectively. $ L(C(X),C(Y))$ is the space of bounded linear operators from $ C(X)$ to $ C(Y)$. If $ E$ is a Banach space, then $ S(E)$ is the closed unit ball in $ E$. An operator $ T$ in $ S(L(C(X),C(Y)))$ is nice if $ {T^ \ast }(\operatorname{ext} S(C{(Y)^ \ast })) \subset \operatorname{ext} S(C{(X)^ \ast })$. For each $ y \in Y,{\varepsilon _y}$ denotes point mass at $ y$. The main theorem states that if $ T$ is extreme in $ S(L(C(X),C(Y)))$ and $ \vert\vert{T^ \ast }({\varepsilon _y})\vert\vert = 1$ for all $ y \in Y$, then $ T$ is nice. Other theorems are proved by using the same techniques as in the proof of the main theorem.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0405173-9
Keywords: Extreme operators, nice operators, compact operator, extremally disconnected, basically disconnected
Article copyright: © Copyright 1976 American Mathematical Society

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