The norm closed triple semigroup algebra

TL;DR

This paper introduces and analyzes a new operator norm-closed triple semigroup algebra, demonstrating its structure as a semi-crossed product, its automorphisms, and its chirality property, extending previous work on the w*-closed algebra.

Contribution

It defines the operator norm-closed triple semigroup algebra $A_{ph}^{G+}$ and characterizes its structure, automorphisms, and chirality, providing new insights into its algebraic and operator-theoretic properties.

Findings

$A_{ph}^{G+}$ is a triple semi-crossed product.

Automorphisms of $A_{ph}^{G+}$ are characterized.

$A_{ph}^{G+}$ is chiral with respect to isometric isomorphisms.

Abstract

The w*-closed triple semigroup algebra was introduced by Power and the author in [19], 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 $A_{ph}^{G+}$ 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 $A_{ph}^{G+}$ is determined and $A_{ph}^{G+}$ is shown to be chiral with respect to isometric isomorphisms.