Wandering vectors and the reflexivity of free semigroup algebras

Matthew Kennedy 1
  • 1 Pure Mathematics Department, University of Waterloo, Waterloo, Ontario, N2L 3G1 Canada


A free semigroup algebra is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for if the set of images of x under words in the generators of is orthonormal.

We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.