The structure of free semigroup algebra (Q2710296)

A free semigroup algebra is the WOT-closed algebra generated by an \(n\)-tuple of isometries with pairwise orthogonal ranges. This paper provides a general structure theorem for all free semigroup algebras which extends results for important special cases in the literature. The structure theoem highlights the importance of the type L representations, which are the repesentations which provide a free semigroup algebra isomorphic to \({\mathfrak L}_n\), which is generated by the left regular representation of the free semigroup on \(n\) letters. Indeed, every free semigroup algebra has a \(2\times 2\) lower triangular form where the first column is a slice of the von Neumann algebra generated by the isometries, and the 22 entry is a type L algebra. The structure of type L algebras is developed in more detail. In particular, every type L representation has a finite ampliation with a spanning set of wandering vectors. An easy application of the structure theorem is a characterization of the radical. With additional work, one obtains a result of Russo-Dye type showing that the convex hull of the isometries in any free semigroup algebra contains the whole open unit ball.