Hello,
The next Banach spaces webinar is on Friday March 12 at 9AM Central time. Please join us at
https://yorku.zoom.us/j/99330056697?pwd=NlBnTERWTGRPbDQyYitnc0k1bTNqZz09https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fyorku.zoom.us%2Fj%2F99330056697%3Fpwd%3DNlBnTERWTGRPbDQyYitnc0k1bTNqZz09&data=04%7C01%7Cbunyamin.sari%40unt.edu%7Cf9a4c8561e094dd4742c08d8e4384c25%7C70de199207c6480fa318a1afcba03983%7C0%7C1%7C637510277741458444%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=%2BBApHdlFkvKANgx30ScPBdyWTz%2ByvDLFjbXbxuAn9uE%3D&reserved=0
(Please note the new zoom link. It shouldn’t ask for a passcode but if it does use Passcode: 036383)
Speaker: Johann Langemets (University of Tartu) Title: A characterization of Banach spaces containing $\ell_1(\kappa)$ via ball-covering properties
Abstract: In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal $\kappa$, a Banach space $X$ contains an isomorphic copy of $\ell_1(\kappa^+)$ if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by $\kappa$ many open balls not containing $\alpha B_X$, where $\alpha\in (0,1)$. We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities. This is a joint work with S. Ciaci and A. Lissitsin.
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