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Family Floer theory is used to construct a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.
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Mohammed Abouzaid, The family Floer functor is faithful. J. Eur. Math. Soc. 19 (2017), no. 7, pp. 2139–2217DOI 10.4171/JEMS/715