On Banach Envelopes and Duals of Lipschitz-free 𝑝-Spaces for 0<𝑝<1

With the aim to better understand the intricate geometry of the class of Lipschitz-free p p -spaces F p ( M ) \mathcal {F}_p(\mathcal {M}) when 0 > p > 1 0>p>1 , in this note we study their Banach envelopes and prove that if 0 > p > 1 0>p>1 and M \mathcal {M} is a metric space then the Banach envelope map of F p ( M ) \mathcal {F}_p(\mathcal {M}) is one-to-one, thus solving in the positive a problem raised by the first author and Kalton [Israel J. Math. 170 (2009), pp. 317–335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of F p ( M ) \mathcal {F}_p(\mathcal {M}) separates the points of F p ( M ) \mathcal {F}_p(\mathcal {M}) .