Group-symmetric holomorphic functions on a Banach space (Q2830651)

The authors take an algebra of holomorphic functions (e.g., \({\mathcal H}_b( E)\), \({\mathcal A}(B_E)\), \({\mathcal H}^\infty(B_E)\)) and a group of linear mappings on \(E\) which leaves \(U\) invariant and consider \({\mathcal H}_G(U)\), the subalgebra of holomorphic functions \(f\) such that \(f\circ \gamma=f\) for all \(\gamma\) in \(G\). Using the Haar measure, they give conditions for the existence of a continuous linear projection from \({\mathcal H}(U)\) onto \({\mathcal H}_G(U)\). They characterise the set of \(G\)-symmetric holomorphic functions for a number of Banach spaces and groups and determine their spectrum. They show that, in many cases, these algebras are small. For example, when \(E\) is \(C([0,1])\) and \(G\) is the composition group of \(C([0,1])\) arising from all homeomorphisms of \([0,1]\), then \({\mathcal H}_G([0,1])\) is equal to all functions \(f: C([0,1])\to \mathbb C\) such that \(f(x)=F(x(0),x(1))\) for some holomorphic function \(F\) on \( \mathbb C^2\). \(G\)-symmetric holomorphic functions on sequence spaces and \(G\)-symmetric holomorphic functions which arise from measure-preserving maps are also characterised.