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.