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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".

We construct a reflexive and unconditionally saturated Banach space X uh such that its dual X * uh is Hereditarily Indecomposable. We also show that every quotient of X uh has a further quotient which is Hereditarily Indecomposable and every quotient of X * uh has a further quotient with an unconditional basis.

Abstract. We provide a characterization of the Banach spaces X with a Schauder basis (en) n∈N which have the property that the dual space X * is naturally isomorphic to the space L diag (X) of diagonal operators with respect to (en) n∈N . We also construct a Hereditarily Indecomposable Banach space X D with a Schauder basis (en) n∈N such that X * D is isometric to L diag (X D ) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L diag (X D ) is of the form T = λI + K, where K is a compact operator.

For every Banach space X with a Schauder basis consider the Banach algebra RI ⊕ K diag (X) of all diagonal operators that are of the form λI + K. We prove that RI ⊕ K diag (X) is a Calkin algbra i.e., there exists a Banach space Y X so that the Calkin algebra of Y X is isomorphic as a Banach algebra to RI ⊕ K diag (X). Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.where the supremum is taken over all disjoint collections of intervals of N, is finite.The spaces J p are quasi-reflexive of order one. It was for observed by A. D. Andrew and W. L. Green in [AG] that J p , after appropriate renorming, becomes a non-unital Banach algebra when endowed with coordinate-wise multiplication with respect to the basis e 1 , e 2 − e 1 , e 3 − e 2 , . . .. We denote the unitization of James space by Re ω ⊕ J p , for 1 < p < ∞.Theorem III. The spaces Re ω ⊕ J p , 1 < p < ∞ are Calkin algebras.S. F. Bellenot, R. G. Haydon, and E. Odell in [BHO] extended the definition of James, based on the unit vector basis of ℓ p , to an arbitrary space X with an unconditional basis to define the space J(X), the "jamesification" of X. The space J(X) is quasi-reflexive of order one whenever X is reflexive. As it so happens this space is a non-unital Banach algebra as well the unitization of which we denote by Re ω ⊕ J(X). Furthermore, a special subspace of J * (X) which we denote by J * (X), and coincides with J(X) * when X does not contain ℓ 1 , is a separable unital Banach algebra.Theorem IV. For any space X with a normalized unconditional basis the spaces Re ω ⊕ J(X) and J * (X) are Calkin algebras.Next we turn our attention to spaces with unconditional bases endowed with coordinate-wise multiplication. We study these spaces themselves as Banach algebras. We do not prove that they are Calkin algebras but they always embed in them as complemented ideals. In what follows "⊕" denotes the direct sum of Banach spaces.Theorem V. Let A be Banach space with a normalized unconditional basis endowed with coordinate-wise multiplication and B be a Banach algebra. In the cases described in the list bellow there exists a Banach space Y so that the Calkin algebra Cal(Y) contains an idealà isomorphic as a Banach algebra to A and a subalgebrã B isomorphic as a Banach algebra to B so that Cal(Y) =à ⊕B.(a) The space B is C(ω), the space of convergent scalar sequences with pointwise multiplication.