On Banach spaces of large density but few operators
by
Piotr Koszmider
Universidade de Sao Paulo

Motivated by a construction by Shelah and Steprans of a Banach space with a basis of type \omega₁ with few operators, we consider the question of the existence of a Banach space of certain class with transfinite basis of type \lambda bigger than \omega₁ on which there are few operators or nearly few operators (these concepts have to be reformulated from the case of \lambda equal to \omega₁ to give nontrivial results for higher types, however in the case of \lambda equal to \omega₁ they are equivalent to Shelah's and Steprans' definition). We obtain various independence results about the existence of Banach spaces with few or nearly few operators with a basis of given cardinal type. For example there are no spaces with few operators and basis of type \omega₃. For \omega₂ it depends on Chang's Conjecture. For nearly few operators the bound of the type of the basis is 2 to \omega₁, however it could be any cardinal. Some open questions and relation with other structures will be mentioned.

http://www.ime.usp.br/~piotr/stronamat.html

Date received: June 19, 2000