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Session 4. Banach Spaces and Operator Theory with Applications
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Operators preserving \(\ell_\infty\) |
Marek Wójtowicz, Uniwersytet Kazimierza Wielkiego, Instytut Matematyki, Poland
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Let \(Y\) be a Banach space, let the space \(\ell_\infty\) be real,
let \(W\) denote the Banach space \(\ell_\infty/c_0\), and let \(Q\)
denote the quotient map \(\ell_\infty \to W\). In 1981 J. Partington
proved there is a topological embedding \(J\) of \(\ell_\infty\)
into \(W\) such that the composition \(QJ\) is an isometry; in
particular, \(Q\) preserves \(\ell_\infty\). By combining this
result with H. Rosenthal's theorem on operators on
\(\ell_\infty(\Gamma)\) we obtain that every injective operator
\(T:\ell_\infty(\Gamma)/c_0(\Gamma) \to Y\) preserves
\(\ell_\infty\). This allows us to show that natural quotient
mappings of some real function spaces preserve \(\ell_\infty\).
I will also review some results on operators preserving other Banach
spaces.
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