On Weighted Hardy Spaces on Complex Semigroups

(S,α) on a complex open Ol'shanskii semigroup S = G Exp (iW), where 1 ≤p≤∞ and α is an absolute value on the involutive semigroup X. For 1 < p < ∞ we prove the existence of an isometric boundary value map H p (S,α) → L p (G) generalizing the corresponding result of Ol'shanskii for p = 2 and α = 1. In the second part we use the fine structure of the space H 2 (S,1) to prove the existence of a bounded holomorphic function on S whose absolute value has a unique maximum in the boudary point 1Β G and therefore complete the proof of the approximation property of the Poisson kernel and the uniqueness of G as a Shilov boundary of S whenever W does not contain affine line.