Analytic structure on the spectrum of the algebra of symmetric analytic functions on $$L_\infty $$

Abstract

We consider the subalgebra \(H_{bs}(L_\infty )\) of analytic functions of bounded type on \(L_\infty [0,1]\) which are symmetric, i.e. invariant, with respect to measurable bijections of [0, 1] that preserve the measure. Our main result is that \(H_{bs}(L_\infty )\) is isomorphic to the algebra of all analytic functions on the strong dual of the space of entire functions on the complex plane \({\mathbb {C}}.\) From this result we deduce that \(H_{bs}(L_\infty )\) is a test algebra for Michael problem about the continuity of complex valued homomorphisms on Frechet algebras.