Stable phase retrieval in arbitrary Banach lattices

A subspace $E$ of a Banach lattice $X$ is said to do phase retrieval if, for any $f, g \in E$, the equality $|f|=|g|$ automatically implies that $f$ and $g$ are scalar multiples of each other (that is, we can recover an element of $E$ from its modulus, up to a unimodular scalar multiple). Further, $E$ does stable phase retrieval (SPR) if there exists a constant $C>0$ so that, for $f$ and $g$ as above, we have $\inf_{|\lambda|=1} \|f - \lambda g\| \leq C \| |f| - |g| \|$ (the recovery described above is "stable’’).
In the talk we establish two results which allow us to describe SPR subspaces of Banach lattices.
(1) In a real Banach lattice $X$, $E$ does SPR if and only it contains no "almost disjoint’’ pairs of elements.
(2) The SPR condition only needs to be verified on "well separated’’ elements of the unit ball.
These results permit us to give examples of SPR subspaces, and also to describe possible SPR subspaces of classical Banach lattices, such as $L_p$ or $C(K)$.
(Joint work with D.Freeman, B.Pineau, and M.Taylor)