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

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 $\Vert q\Vert =1$ 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 $C^{\ast }$-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.

Arkiv för Matematik, To appear