Wandering vectors and the reflexivity of free semigroup algebras

TL;DR

This paper proves that free semigroup algebras either have a wandering vector or are von Neumann algebras, leading to their reflexivity and hyper-reflexivity with small constants, advancing understanding of their structure.

Contribution

It establishes a dichotomy for free semigroup algebras, showing they either possess a wandering vector or are von Neumann algebras, and proves their reflexivity and hyper-reflexivity.

Findings

Every free semigroup algebra is reflexive.

Certain free semigroup algebras are hyper-reflexive.

A dichotomy: having a wandering vector or being a von Neumann algebra.

Abstract

A free semigroup algebra S 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 S if the set of images of x under non-commuting words in the generators of S 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.