Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

A set in a domain in has the norm-preserving extension property if every bounded holomorphic function on has a holomorphic extension to with the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton, itself, a complex geodesic of , or the union of the set and a complex geodesic of degree in . We also prove that the complex geodesics in coincide with the nontrivial holomorphic retracts in . Thus, in contrast to the case of the ball or the bidisc, there are sets in which have the norm-preserving extension property but are not holomorphic retracts of . In the course of the proof we obtain a detailed classification of the complex geodesics in modulo automorphisms of . We give applications to von Neumann-type inequalities for -contractions (that is, commuting pairs of operators for which the closure of is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.