0000000000733093
AUTHOR
T. V. Vasylyshyn
The algebra of symmetric analytic functions on L∞
We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.
Symmetric and finitely symmetric polynomials on the spaces ℓ∞ and L∞[0,+∞)
We consider on the space l∞ polynomials that are invariant regarding permutations of the sequence variable or regarding finite permutations. Accordingly, they are trivial or factor through c0. The analogous study, with analogous results, is carried out on L∞[0,+∞), replacing the permutations of N by measurable bijections of [0,+∞) that preserve the Lebesgue measure.