Isometries on extremely non-complex Banach spaces

Abstract

Abstract. Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that Id +T 2 = 1 + T 2 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.

Decompositions of Banach spaces C(K) with few operators

“…The reverse is not true, as it is proved in [24]. However, the implication hold when we add some properties to K : It is proved in [20] that C(K ) satisfying the condition stated in (3) or (4) also satisfies the one stated in (5), providing K is perfect, i.e, it does not contain isolated points. All the constructions made in the cited references of Koszmider or weakly Koszmider spaces are perfect.…”

“…In [2] it is proved the existence of a compactum K satisfying (E) but none of the items (A) to (D). Another construction with not few operators, in the sense of item (A) or (B), but still with properties related to a small space of operators, were made in [20]. This is how this paper is organized.…”

Algebraic and invariance properties of the group of isometries

“…A Banach space X is said to be extremely noncomplex if Id + T 2 = 1 + T 2 for every T ∈ B(X) [23]. In that paper the authors show that G(X) is a discrete Boolean group whenever X is extremely noncomplex.…”

A C(K) Banach space which does not have the Schroeder–Bernstein property

“…This approximation is developed in Section 5 which relies on Section 4 where we prove general properties of operators on X + . Some of the ideas of Section 4 which we use for dealing with compactifications of disjoint K i s where for each i ∈ N the space C(K i ) has few operators were developed in [17] and [16].…”