Hello,
The next Banach spaces webinar is on Friday July 16 at 9AM Central time. Please join us at
https://unt.zoom.us/j/83807914306
Title: Compact retractions and the $\pi$-property of Banach spaces Speaker: Rubén Medina (Granada)
Abstract: In the talk we will focus on Lipschitz retractions of a separable Banach space $X$ onto its closed and convex generating subsets $K$, a question asked by Godefroy and Ozawa in 2014. Our results are concerning the case when $K$ is in some quantitative sense small, namely when $K$ is in very little neibourhoods of certain finite dimensional sections of it. Under such assumptions we obtain a near characterization of the $\pi$-property (resp. Finite Dimensional Decomposition property) of a separable Banach space $X$. In one direction, if $X$ admits the Finite Dimensional Decomposition (which is isomorphically equivalent to the metric-$\pi$-property) then we will see how to construct a Lipschitz retraction onto a (small) generating convex compact $K$. On the other hand, we will prove that if $X$ admits a small (in a precise sense) generating compact Lipschitz retract then $X$ has the $\pi$-property. It seems to be an open problem whether the $\pi$-property is isomorphically equivalent to the metric-$\pi$-property (a positive answer would turn our results into a complete characterization). In the case of dual Banach spaces, this characterization is indeed valid.
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