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

The weak operator topology closed operator algebra on L2(R)L2(R) generated by the one-parameter semigroups for translation, dilation and multiplication by eiλx,λ≥0eiλx,λ≥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, AphAph, is antisymmetric in the sense that Aph∩Aph⁎=CI, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is RR. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with AphAph and Aph⁎ being chiral representatives.