Gonzalo Martinez Cervantes: L-orthogonal sequences versus L-orthogonal elements
25/03/21 02:47
Tuesday, March 30, 2021 17:00
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Gonzalo Martinez Cervantes (University of Murcia)
Title: L-orthogonal sequences versus L-orthogonal elements
Abstract: Let \(X\) be a Banach space. We say that a sequence \(\{x_n\}_n\) in the sphere of a Banach space \(X\) is an L-orthogonal sequence if the norm of \(x+x_n\) converges to \(1+\|x\|\) for every \(x\) in \(X\). On the other hand, we say that an element \(x^{**}\) in the sphere of \(X^{**}\) is L-orthogonal to \(X\) if the norm of \(x^{**}+x\) is equal to \(1+\|x\|\) for every \(x\) in \(X\). In this talk we will recall some results due to G. Godefroy, N. J. Kalton, B. Maurey, V. Kadets, V. Shepelska and D.Werner relating these concepts to the containment of an isomorphic copy of \(\ell_1\). It is natural to conjecture that the weak*-closure of an L-orthogonal sequence always contains L-orthogonal elements in the bidual. Indeed, this is the case for separable Banach spaces. We will see that this conjecture is independent of ZFC. Namely, we provide an affirmative answer under the existence of selective ultrafilters, whereas a counterexample can be constructed if no Q-point exists.
This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Gonzalo Martinez Cervantes (University of Murcia)
Title: L-orthogonal sequences versus L-orthogonal elements
Abstract: Let \(X\) be a Banach space. We say that a sequence \(\{x_n\}_n\) in the sphere of a Banach space \(X\) is an L-orthogonal sequence if the norm of \(x+x_n\) converges to \(1+\|x\|\) for every \(x\) in \(X\). On the other hand, we say that an element \(x^{**}\) in the sphere of \(X^{**}\) is L-orthogonal to \(X\) if the norm of \(x^{**}+x\) is equal to \(1+\|x\|\) for every \(x\) in \(X\). In this talk we will recall some results due to G. Godefroy, N. J. Kalton, B. Maurey, V. Kadets, V. Shepelska and D.Werner relating these concepts to the containment of an isomorphic copy of \(\ell_1\). It is natural to conjecture that the weak*-closure of an L-orthogonal sequence always contains L-orthogonal elements in the bidual. Indeed, this is the case for separable Banach spaces. We will see that this conjecture is independent of ZFC. Namely, we provide an affirmative answer under the existence of selective ultrafilters, whereas a counterexample can be constructed if no Q-point exists.
This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.