Some examples of extremal triples of commuting contractions

TL;DR

This paper investigates extremal triples of commuting contractions, showing that many known non-extendable triples are actually extremal, and explores their model structure within the family of all such triples.

Contribution

It characterizes extremal triples in the family of commuting contractions and demonstrates that many known non-extendable examples are extremal.

Findings

Many triples without coisometric extensions are extremal.

Extremal elements generate an optimal model for the family of triples.

The paper provides examples and structural insights into extremal triples.

Abstract

The collection $\frk{C}_3$ of all triples of commuting contractions forms a family in the sense of Agler, and so has an "optimal" model $\pd\frk{C}_3$ generated by its extremal elements. A given $T\in\frk{C}_3$ is extremal if every $X\in\frk{C}_3$ extending $T$ is an extension by direct sum. We show that many of the known examples of triples in $\frk{C}_3$ that fail to have coisometric extensions are in fact extremal.