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The Hubbard model in the limit of infinite $U$ is investigated within aprojected slave fermion representation and following a previous work of theauthor and collaborators \cite{BFK}. The stability of the Nagaoka's phase withrespect to a non vanishing concentration of holes ($\delta_h$) is analyzed byenvisaging the existence of a spin effective action for itinerant magnetism ofthe Hubbard model. It is considered that,as the hole doping increases away fromthe half filled insulating limit, free holes are expected to be moredelocalized. Depending on treatment for the hopping: a ferromagnetic oranti-ferromagnetic ordering might arise and the Nagaoka's phase might have somestability with respect to $\delta_h \neq 0$.

Stability of Nagaoka phase, spin effective action and delocalized free holes