  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.
