© 1999 by London Mathematical Society
Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras
Department of Pure Mathematics, University of Waterloo Waterloo, Ontario N2L3G1, Canada. E-mail: krdavidson{at}math.uwaterloo.ca
Department of Mathematics, University of Nebraska Lincoln, NE68588, U.S.A. E-mail: dpitts{at}math.unl.edu
Received 19 August 1997.
A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on n generators. This algebra is the appropriate non-commutative n-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51.
The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras On. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive. 1991 Mathematics Subject Classification: 47D25.