A note on absolute continuity in free semigroup algebras

A free semigroup algebra is the weak operator topology closed (nonself-adjoint, unital) algebra generated by n isometries with pairwise orthogonal ranges. The prototype is the algebra generated by the left regular representation of the free semigroup on n letters. A free semigroup algebra which is isomorphic to the left regular algebra is called type L. If the infinite ampliation of the isometries generates a type L algebra, it is called weak-* type L. A free semigroup algebra is absolutely continuous if the vector functionals on it are equivalent to (some) vector functionals on the left regular representation. The purpose of this note is to show that absolutely continuous free semigroup algebras are weak-* type L