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 $exp(i\lambda x),\lambda \ge 0$, 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 $A_{ph}\cap A_{ph}^{\ast }=CI$, 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 $A_{ph}^{\ast }$ being chiral representatives.

A proof has been corrected and a clarifying figure added (Figure 4). New results have been added (Corollary 5.4 and Theorem 5.7). Minor corrections and changes of order have been made