Geometry of unit balls of free Banach lattices, and its applications

We begin by describing the unit ball of the free $p$-convex Banach lattice over a Banach space $E$ (denoted by ${\mathrm{F}\mathrm{B}\mathrm{L}}^{(p)}[E]$) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space $E$ has the $\lambda$-Approximation Property, then ${\mathrm{F}\mathrm{B}\mathrm{L}}^{(p)}[E]$ has the $\lambda$-Positive Approximation Property. Further, we show that operators $u\in B(E,F)$ (where $E$ and $F$ are Banach spaces) which extend to lattice homomorphisms from ${\mathrm{F}\mathrm{B}\mathrm{L}}^{(q)}[E]$ to ${\mathrm{F}\mathrm{B}\mathrm{L}}^{(p)}[F]$ are precisely those whose adjoints are $(q,p)$-mixing. Related results are also obtained for free lattices with an upper $p$-estimate.