Let f:D2 → C be an analytic (= holomorphic) function of 2 complex variables which has the additional property of being symmetric (i.e. f(z1, z2) = f(z2, z1), for all (z1, z2) ∈ D2). We study properties of the bidisc algebra of such functions, As(D2). In particular, we investigate questions related to polynomial convexity as well as to the maximal ideal space of As(D2). (This is related to work of J. Agler and N. Young on the symmetrized bidisc [1].) In addition, we examine similar algebras when D2 is replaced by another symmetric subset of Cn. Finally, we extend the discussion to symmetric holomorphic functions on C(K), for compact sets K.

This talk reports on work in progress with Javier Falcó and Manuel Maestre [3], which in turn builds on earlier work in [2, 4].