Some examples of extremal triples of commuting contractions

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.