We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 25 June 2025
We discuss the relationships between a certain class of uniform algebras, called tight uniform algebras, and various concepts from Banach space theory, such as the Dunford—Pettis property, the Pelczynski property, and weak sequential completeness. We also mention some connections with the ö-problem, interpolation, pointwise bounded approximation, and inner functions on strictly pseudoconvex domains.
1. Introduction
B. Cole and T. W. Gamelin [1982] introduced a generalized notion of analyticity, which they called tightness. If K is a compact space and X ⊂ C(K) is a closed subspace (in the uniform norm) we say X is a tight subspace if the Hankeltype operator Sg : X → C(K)/X defined by / i→ fg + X is weakly compact for every g ∈ C(K). Recall that a uniform algebra A on K is a closed, separating subalgebra of C(K) which contains the constants. We say a uniform algebra A on K is a tight uniform algebra if it is a tight subspace of C(K). The following prototypical example from [Cole and Gamelin 1982] illustrates how tightness could be thought of as an abstract version of the solvability of a -problem with a mild gain in smoothness.
To save this book to your Kindle, first ensure no-reply@cambridge-org.demo.remotlog.com 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.
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.
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.
