Dear all,

The next Banach spaces webinar is on Friday July 31 9AM CDT (e.g., Dallas, TX time). Please join us at

https://unt.zoom.us/j/512907580

Speaker: Valentin Ferenczi (University of São Paulo)

Title: On envelopes and L_p​-spaces

Abstract. This talk is based on a work in progress with Jordi Lopez-Abad.

We define, inside a given space $X$, the {\em envelope} ${\rm Env}(Y)$ of a subspace $Y$ as the largest subspace such that, for any net of surjective isometries on $X$, pointwise convergence to the identity on $Y$ implies pointwise convergence to the identity on ${\rm Env}(Y)$. This is reminiscent of the study of Korovkin sets in the spaces $C(0,1)$ or $L_p(\mu)$ (initiated by P.P. Korovkin in 1960).

We shall mention some results of a recent paper of J. Lopez-Abad, B. Mbombo, and S. Todorcevic and myself (2019): different notions of ultrahomogeneity of Banach spaces will be stated (AUH, Fra"iss'e) which are relevant to multidimensional versions of Mazur rotations problem. Known examples of these are the Gurarij space and the spaces $L_p$'s for $p \neq 4,6,8,\ldots$. We shall address the conjecture that these are the only separable examples.

The notion of envelope is especially relevant to the study of AUH or Fra"iss'e spaces. In particular we shall compute explicitely certain envelopes in $L_p$-spaces and conclude by giving a meaning to potentially new objects such as $L_p/\ell_2$, $L_p/L_q$, $L_p/\ell_q$, for appropriate values of $p$ and $q$.

Partially supported by Fapesp, 2016/25574-8 and CNPq, 303731/2019-2.

* For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach

Upcoming schedule

August 7: Pete Casazza (University of Missouri)

Thank you, and best regards,

Bunyamin Sari