Free Semigroup Algebras a Survey

Free semigroup algebras are wot-closed algebras generated by n isometries with pairwise orthogonal ranges. They were introduced in [27] as an interesting class of operator algebras in their own right. The prototype algebra, obtained from the left regular representation of the free semigroup on n letters, was introduced by Popescu [45] in connection with multi-variable non-commutative dilation theory. This algebra has a great deal of analytic structure associated to the unit ball in Cn which justifies its name as the noncommutative analytic Toeplitz algebra. The general free semigroup algebras contain interesting computable information about the unitary invariants for the n-tuple of generators. This has allowed the classification of large classes of representations of the Cuntz algebra. Such classifications are important in various applications of C*-algebras. In particular, the work of Bratteli and Jorgensen [15] uses such representations to generate wavelets, and unitary invariants for special classes of representations are central to their work. In this article, we will survey results about free semigroup algebras themselves, with passing reference to various applications.