The operator algebra generated by the translation, dilation and multiplication semigroups

TL;DR

This paper characterizes the operator algebra generated by translation, dilation, and multiplication semigroups on L^2(R), revealing its reflexivity, invariant subspace structure, and automorphism group, with implications for operator algebra theory.

Contribution

It provides a detailed analysis of the structure and properties of the triple semigroup operator algebra, including reflexivity, antisymmetry, ideals, and automorphisms, and classifies its unitary equivalence classes.

Findings

The algebra is reflexive with a binest invariant subspace lattice.

It is antisymmetric with a nonzero proper weakly closed ideal.

The automorphism group of the algebra is isomorphic to R.

Abstract

The weak operator topology closed operator algebra on $L^2(R)$ generated by the one-parameter semigroups for translation, dilation and multiplication by $exp(i\lambda x), \lambda \geq 0$, is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, $A_{ph}$, is antisymmetric in the sense that $A_{ph} \cap A_{ph}^*= CI$, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is $R$. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with $A_{ph}$ and $A_{ph}^*$ being chiral representatives.