The w*-closed triple semigroup algebra was introduced by Power and the author in [21], where it was proved to be reflexive and to be chiral, in the sense of not being unitarily equivalent to its adjoint algebra. Here an analogous operator norm-closed triple semigroup algebra AphG+ is considered and shown to be a triple semi-crossed product for the action on analytic almost periodic functions by the semigroups of one-sided translations and one-sided dilations. The structure of isometric automorphisms of AphG+ is determined and AphG+ is shown to be chiral with respect to isometric isomorphisms.