Completely almost periodic functionals

TL;DR

This paper introduces the concept of completely almost periodic functionals on completely contractive Banach algebras and demonstrates their structure in the context of injective Hopf--von Neumann algebras.

Contribution

It defines completely almost periodic functionals using complete compactness and shows their algebraic structure in specific operator algebra settings.

Findings

The space of completely almost periodic functionals forms a C*-subalgebra in certain Hopf--von Neumann algebras.

The notion of complete compactness is key to understanding these functionals.

The paper extends classical concepts to the setting of operator algebras.

Abstract

Using the notion of complete compactness introduced by H. Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if $(M,\Gamma)$ is a Hopf--von Neumann algebra with $M$ injective, then the space of completely almost periodic functionals on $M_\ast$ is a $C^\ast$-subalgebra of $M$.