Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-04T04:44:17.829Z Has data issue: false hasContentIssue false

8 - Extension of C(K)-Valued Operators

Published online by Cambridge University Press:  19 January 2023

Félix Cabello Sánchez
Affiliation:
Universidad de Extremadura, Spain
Jesús M. F. Castillo
Affiliation:
Universidad de Extremadura, Spain
Get access

Summary

The chapter is devoted to the single topic of extending $\mathscr C$-valued operators. Its first section presents the global approach to the extension of operators: Zippin’s characterisation of $\mathscr C$-trivial embeddings by means of weak*-continuous selectors and a few noteworthy applications. The second section presents the Lindenstrauss-Pe\l czy\’nski theorem with two different proofs: the first one combines homological techniques with the global approach, while the second is Lindenstrauss-Pe\l czy\’nski’s original proof. The analysis of their proof is indispensable for understanding Kalton’s imaginative inventions that lead to the so-called $L^*$ and $m_1$-type properties and to a decent list of $\mathscr C$-extensible spaces. The next two sections contain, respectively, those points of the Lipschitz theory that are necessary to develop the linear theory and different aspects of Zippin’s problem: which separable Banach spaces $X$ satisfy $\operatorname{Ext}(X, C(K))=0\,$? The problem admits an interesting gradation in terms of the topological complexity of $K$. The final section reports the complete solution of the problem of whether $\operatorname{Ext}(C(K), c_0)\neq 0$ for all non-metrisable compacta $K$.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×