| Grant number: | 12/25171-0 |
| Support Opportunities: | Scholarships in Brazil - Doctorate |
| Start date: | May 01, 2013 |
| End date: | July 31, 2014 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Analysis |
| Principal Investigator: | Daniel Victor Tausk |
| Grantee: | Claudia Correa de Andrade Oliveira |
| Host Institution: | Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil |
Abstract Banach spaces of the form C(K) (the space of real-valued continuous functions defined on the compact Hausdorff space K, endowed with the supremum norm) have a very important role in the general theory of Banach spaces. For instance, several questions concerning Banach spaces are answered in the negative through counter-examples in the class of C(K) spaces.In this project we intend to study, in the class of C(K) spaces, some classical problems of the general theory of Banach spaces, such as the subspace complementation problem, the problem of extension of bounded operators, and the problem of existence of copies (isomorphic or isometric, complemented or not) of classical sequence spaces inside Banach spaces. Recall that Banach spaces of the form C(K) have a natural Banach algebra structure; therefore, among the closed subspaces of a C(K) space we can consider, in particular the Banach subalgebras.Regarding the subject of extension of bounded operators, a type of question that we wish to investigate is the following: how to characterize the class of compact Hausdorff spaces K such that every c_0-valued bounded operator defined in a closed subspace of C(K) admits a bounded extension to the entire space C(K)?Recall that the celebrated Theorem of Sobczyk states that the space c_0 is separably injective, i.e., given a separable Banach space X then every bounded c_0-valued operator defined in a closed subspace of X admits a bounded extension to the entire space X. Thus Sobczyk's theorem implies that if K is metrizable (equivalently, if C(K) is separable) then K belongs to the class of compact spaces considered in the question stated above. Such question is just one among many that we would like to investigate in the quest for generalizations of Sobczyk's theorem. The quest for such generalizations is a rich field of investigation in which a lot of work has been done in the last decades. The advisor and the candidate have submitted for publication in november 2012 an article on this subject. | |
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