The Grothendieck property for spaces $Lip_0(M)$ of Lipschitz functions

Jerzy KÄ…kol (Adam Mickiewicz University in Poznan)

Mar 19. 2025, 11:45 — 12:05

We study the following
\begin{problem}
For which metric spaces $M$, in particular for which Banach spaces $M$, do the spaces $Lip_0(M)$ have the Grothendieck property?
\end{problem}
Recall that a Banach space $E$ is called \emph{Grothendieck} if every weak* convergent sequence in the dual space $E^*$  converges weakly. Typical examples of Grothendieck spaces are reflexive spaces, the space $\ell_\infty$ or more generally spaces $C(K)$ for $K$ extremally disconnected, the space $H^\infty$ of all bounded analytic functions on the unit disk and von Neumann algebras.  It seems that apart from $Lip_0([0,1])\simeq\ell_\infty$ there is no known example of a Banach space  $Lip_0(M)$ which is a Grothendieck space. We provide a number of conditions for metric spaces $M$ implying that the corresponding  spaces $Lip_0(M)$ are not Grothendieck. For example, if a Banach space $E$ is a $C(K)$-space, $L_1(\mu)$-space, $Lip_0(M)$-space, or $\mathcal{F}(M)$-space, then $Lip_0(E)$ is  not Grothendieck.

The presented results  are   based on a joint work with Christian  Bargetz and Damian Sobota.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Structures in Banach Spaces (Workshop)
Organizer(s):
Antonio Aviles (U Murcia)
Vera Fischer (U of Vienna)
Grzegorz Plebanek (U of Wroclaw)
Damian Sobota (U of Vienna)