Related papers: An elementary example of a Banach space not isomor…
An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.
There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the…
We give examples of real Banach spaces with exactly infinite countably many complex structures and with $\omega_1$ many complex structures.
A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to…
Necessary and sufficient conditions for a separable Banach space to be a dual space are proved. Some applications are discussed
There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and…
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…
We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples…
We prove that there is a compact space $L$ and a 1-complemented subspace of the Banach space $C(L)$ which is not isomorphic to a space of continuous functions.
We construct a reflexive Banach space $X$ with a subspace isometric to $X$, which is not complemented in $X$.
This article is a continuation of a paper of the first author \cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI…
A Banach space contains asymptotically isometric copies of $\ell_1$ if and only if its dual space contains an isometric copy of $L_1$.
Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and…
It is shown how to construct, given a Banach space which does not have the approximation property, another Banach space which does not have the approximation property but which does have the compact approximation property.
Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin…
If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0 and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using the…
We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in the previous paper Banach spaces without minimal subspaces, by classifying them according to which side of the…
It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism…
We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces
In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: If X is a real Banach space…