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

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.