$\Lambda$ polarizing fragmentation function from Belle $e^+e^-$ data

Experimental data from Belle Collaboration for the transverse polarization of $\Lambda$'s measured in $e^+ e^-$ annihilation processes are used to extract the polarizing fragmentation function (FF) within a TMD approach. We consider both associated and inclusive $\Lambda$ production, showing a quite consistent scenario. Good separation in flavor is obtained, leading to four independent FFs. Predictions for SIDIS processes at the EIC, crucial for understanding their universality and evolution properties, are also presented.