This is joint work with J. Lopez-Abad† and S. Todorcevic‡.

We present a reflexive Banach space Xω1 with a transfinite basis. (eα)α<ω1 with the following properties, among others:

1. There is no unconditional basic sequence in Xω1.

2. Each block sequence (xn)n<ω relative to the basis generates an Hereditarily Indecomposable (HI) space.

3. Each operator T ∈ L(Xω1) is of the form T = Tf + S, where S is a strictly singular operator and where Tf is defined by Tf (eα) = f(α)eα for certain continuous function f : ω1 → R.

4. Xω1 is not isomorphic to any proper subspace.

5. Xω1 is not isomorphic to any non trivial quotient.

6. The only projections are of the form PI1 + · · · + PIn + S, where Ii are intervals of ordinals, PIi are the natural projections associated to Ii, and S is strictly singular.

7. The basis (eα)α<ω1 is nearly spreading. In particular, (en)n<ω is a nearly spreading basis1 for the reflexive HI space henin<ω, which makes the space considerably different from the HI space by Gowers-Maurey.