Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra

Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydisk Δ ¯ ∞ generated by the coordinate functions z1 z2,…. Let σ be the Haar measure on the infinite torus T∞, which is the distinguished boundary of the infinite polydisk. For fixed p in the range 1 < p, < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure in LP(σ) if and only if ζ ∈ l2. A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 < p < ∞, is that the point evaluation at ζ is continuous in the Lp-norm if and only if ζ ∈ Δ∞ ∩l2. In this case the functions in Hp are shown to correspond to analytic functions on the domain Δ∞ l2 in l2. Along these same lines, it is shown that H∞ (σ) is isometrically isomorphic to H (Δ∞ ∩ l2), and also to H∞(B), where B is the open unit ball of the sequence space c0.