The Hopf structure of some dual operator algebras

TL;DR

This paper explores the Hopf algebra structures of certain dual operator algebras linked to semigroups, including the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury-Arveson space, revealing their dual Hopf algebra properties.

Contribution

It introduces the framework of dual Hopf algebra structures for these operator algebras and analyzes their properties within dilation theory.

Findings

The algebras and their preduals form dual Hopf algebras.

Structural results about non-self-adjoint dual Hopf algebras.

Connections to dilation theory and semigroup representations.

Abstract

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury-Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.