Operator algebras from the discrete Heisenberg semigroup

TL;DR

This paper investigates the structure and reflexivity of operator algebras generated by representations of the discrete Heisenberg semigroup, revealing conditions for reflexivity and providing examples of non-reflexive cases.

Contribution

It demonstrates that the left regular representation yields a semi-simple reflexive algebra and provides an example of a non-reflexive algebra, advancing understanding of operator algebra structures.

Findings

Left regular representation produces a semi-simple reflexive algebra

An example of a non-reflexive algebra from a different representation

Reflexivity results for subspaces of $H^{e}(b T) ensor b B(b H)$

Abstract

We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{\infty}(\bb{T})\otimes\cl B(\cl H)$.