Hello, The next Banach spaces webinar is on Friday March 5 at 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Speaker: Antonio Avilés López, Universidad de Murcia Title: Sequential octahedrality and L-orthogonal elements Abstract: Given a Banach space $X$, we consider the following two isometric properties, variations on the notion of octahedrality that can be traced back to the work of B. Maurey: 1. There is an element $e^{**}$ in the sphere of the bidual such that $\|e^{**}+x\| = 1 + \|x\|$ for every $x\in X$. 2. There is a sequence $(e_n)$ in the sphere of $X$ such that $\lim_n \|e_n+x\| = 1 + \|x\|$ Uncountable sums provide examples that 1 does not imply 2. But the converse is unclear. It is natural to conjecture that a weak$^*$-cluster point of the sequence $(e_n)$ would give the desired $e^{**}$. This turns out to be independent of the usual axioms of set theory. The proof involves understanding different kinds of ultrafilters that may or may not exist, as well as a filter version of the Lebesgue dominated convergence theorem, similar to those considered by V. Kadets and A. Leonov. This is a joint work (in progress) with G. Mart\'{\i}nez Cervantes and A. Rueda Zoca. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari