Javier Merí

Slicely countably determined Banach spaces

We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon-Nikodým property and all spaces without copies of l 1 . We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach space with the alternative Daugavet property contains l 1 and that operators which do not fix copies of l 1 on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.

EXTREMELY NON-COMPLEX C(K) SPACES

Abstract We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality ‖ Id + T 2 ‖ = 1 + ‖ T 2 ‖ holds for every bounded linear operator T : X → X . This answers in the positive Question 4.11 of [V. Kadets, M. Martin, J. Meri, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C ( K ) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C ( K ) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K 1 and K 2 such that C ( K 1 ) and C ( K 2 ) are extremely non-complex, C ( K 1 ) contains a complemented copy of C ( 2 ω ) and C ( K 2 ) contains a (1-complemented) isometric copy of l ∞ .

Isometries on extremely non-complex Banach spaces

Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

ON THE NUMERICAL INDEX OF REAL Lp(µ)-SPACES

We give a lower bound for the numerical index of the real space Lp(µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space Lp(µ), one has

Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces

\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||

Lipschitz slices and the Daugavet equation for Lipschitz operators

where

A note on the numerical index of the Lp space of dimension two

{M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0