Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality

TL;DR

This paper extends the theory of dilations for $q$-commuting contractions, establishing conditions for regular $q$-unitary dilations, and proves a von Neumann type inequality for these families.

Contribution

It generalizes the dilation theorem to $q$-commuting contractions with $ orm{q}=1$, linking Brehmer's positivity to $Q$-unitary dilations.

Findings

Equivalence of dilation conditions for $q$-commuting contractions.

Application of Stinespring's and Naimark's theorems in the proof.

Identification of cases where $q$-commuting contractions admit dilations.

Abstract

It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to any family $\mathcal T$ of $q$-commuting contractions with $\|q\|=1$ by showing the equivalence of the following three statements: $(i)$ $\mathcal T$ admits a regular $q$-unitary dilation; $(ii)$ $\mathcal T$ satisfies Brehmer's positivity condition; $(iii)$ $\mathcal T$ admits a $Q$-unitary dilation for a family of $Q$-commuting unitaries. We achieve the first part of the result by an application of Stinespring's dilation theorem on a particular completely positive map acting on a quotient algebra of a group $C^*$-algebra, where the underlying group is a free group, and the second part is obtained by an application of Naimark's theorem. Next, we find several cases when $\mathcal{T}$ admits a regular $q$-unitary dilation and establish a von Neumann type inequality for such a $q$-commuting family.