The Structure of Free Semigroup Algebras

A free semigroup algebra is the wot-closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The interest in these algebras arises primarily from two of their interesting features. The first is that they provide useful information about unitary invariants of representations of the Cuntz–Toeplitz algebras. The second is that they form a class of nonself-adjoint operator algebras which are of interest in their own right. This class contains a distinguished representative, the “non-commutative Toeplitz algebra,” which is generated by the left regular representation of the free semigroup on n letters and denoted Ln. This paper provides a general structure theorem for all free semigroup algebras, Theorem 2.6, which extends results for important special cases in the literature. The structure theorem highlights the importance of the type L representations, which are the representations which provide a free semigroup algebra isomorphic to Ln. Indeed, every free semigroup algebra has a 2 × 2 lower triangular form where the first column is a slice of a von Neumann algebra and the 22 entry is a type L algebra. We develop the structure of type L algebras in more detail. In particular, we show in Corollary 1.9 that every type L representation has a finite ampliation with a spanning set of wandering vectors. As an application of our structure theorem, we are immediately able to characterize the radical in Corollary 2.9. With additional work, we obtain Theorem 4.5 of Russo–Dye type showing that the convex hull of the isometries in any free semigroup algebra contains the whole open unit ball. Finally we obtain some information about invariant subspaces and hyper-reflexivity.