Additive units of product systems

TL;DR

This paper introduces additive units ('addits') in pointed Arveson systems, revealing their structure, independence from reference units, and applications in amalgamation and cluster constructions, advancing the understanding of product systems in operator algebra.

Contribution

It defines addits in pointed Arveson systems, explores their properties, and applies them to develop new results in amalgamation and subsystem analysis.

Findings

Addits form a Hilbert space with roots isomorphic to the index space.

Addits generate the type I part of the Arveson system.

Explicit formula for amalgamation of pointed Arveson systems.

Abstract

We introduce the notion of additive units, or `addits', of a pointed Arveson system, and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of `roots' is isomorphic to the index space of the Arveson system, and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology `spatial product' of spatial Arveson systems). Finally a cluster construction for inclusion subsystems of an Arveson system is introduced and we demonstrate its correspondence with the action of the Cantor--Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher.