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 T of q-commuting contractions with \|q\|=1 by showing the equivalence of the following three statements: (i) T admits a regular q-unitary dilation; (ii) T satisfies Brehmer's positivity condition; (iii) 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 T admits a regular q-unitary dilation and establish a von Neumann type inequality for such a q-commuting family.