Symmetric Analytic Functions on Banach Spaces Associated with the Cantor Set

We consider Banach spaces ℓp(C),1≤p<∞, where the index set C is the classical Cantor set and study various groups of symmetries of ℓp(C), associated with the binary representation of C. The main purpose of the paper is the investigation of polynomials on ℓp(C) that are symmetric (i.e., invariant) with respect to the constructed groups G. We are interested in finding systems of generators of algebras of G-symmetric polynomials for different groups G and we discuss possible applications of G-symmetric polynomials to highly composite physical systems. The generators are useful for descriptions of spectra of algebras of G-symmetric analytic functions on ℓp(C), and for the construction of some nontrivial complex homomorphisms of these algebras. Finally, we establish the topological transitivity and hypercyclicity of some shift-like operators on ℓp(C) and its subspaces, and translation operators on algebras of symmetric analytic functions on ℓp(C).