Contents 0. Introduction 1 0.1. Hereditarily James Tree spaces 3 0.2. Saturated extensions 7 0.3. The attractors method 10 1. Strictly singular extensions with attractors 15 2. Strongly strictly singular extensions 24 3. The James tree space JT F2 . 32 4. The space (X F2 ) * and the space of the operators L((X F2 ) * ) 40 5. The structure of X * F2 and a variant of X F2 46 6. A nonseparable HI space with no reflexive subspace 52 7. A HJT space with unconditionally and reflexively saturated dual 63 Appendix A. The auxiliary space and the basic inequality 68 Appendix B. The James tree spaces JT F2,s , JT F2 and JT Fs 75 References 86 Research partially supported by EPEAEK program "PYTHAGORAS".

Abstract. The class of quasi Radon-Nikodým compact spaces is introduced. We prove that this class is closed under countable products and continuous images. It includes the Radon-Nikodým compact spaces. Adapting Alster's proof we show that every quasi Radon-Nikodým and Corson compact space is Eberlein. This generalizes earlier results by J. Orihuela, W. Schachermayer, M. Valdivia and C. Stegall. Further the class of almost totally disconnected spaces is defined and it is shown that every quasi Radon-Nikodým space which is almost totally disconnected is actually a Radon-Nikodým compact space embeddable in the space of probability measures on a scattered compact space.

Abstract. We introduce the notion of J-continuity, which generalizes both continuity and the hypothesis in the Generalized Banach Contraction Conjecture, and prove that any J-continuous self-map on a scattered compact space, has an invariant finite set. We use the results and the techniques to prove the Generalized Banach Contraction Conjecture.

Abstract. We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set C to [0, 1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from C to [0, 1] is dense in the supremum norm in the set of all continuous surjections. The nonmetrizable case is also investigated. As a consequence, we obtain a new characterization of Eberlein compact sets.

Abstract. In the present paper we provide sufficient conditions such that a normalized pointwise convergent to zero sequence in C(K, X) with K a compact space and X a Banach space has an unconditional subsequence.As a consequence we obtain that any such sequence of functions (f n ) n with finite and uniformly bounded cardinality of their range admits an unconditional subsequence.