Unitary dilation of freely independent contractions

Abstract

Abstract. Inspired by the Sz.-Nagy-Foias dilation theorem we show that n freely independent contractions dilate to n freely independent unitaries. IntroductionThe Sz.-Nagy-Foias dilation theorem is a celebrated result in classical dilation theory. It says that n doubly commuting contractions can be simultaneously dilated to n doubly commuting unitaries. This was the original multivariable dilation theory context proven by Brehmer and Sz.-Nagy [7,25,26] until Andô [1] proved that one can do this for just commuting and not doubly commuting contractions when n = 2. However, it was subsequently shown in [20] and [27] that there are three commuting contractions which do not dilate to three commuting unitaries. This obstruction spurred on dilation theories in other contexts [2,8,11,14,23] and many other generalizations. One recent usage of dilations of doubly commuting contractions is the dilation of Nica covariant representations of lattice-ordered semigroups [15,17].Doubly commuting is one of two ingredients in the notion of tensor independence (or classical independence). It is natural then to ask whether n tensor independent contractions can be dilated to n tensor independent unitaries. The answer is yes (Theorem 2.2) and begs the question whether this can be done with other notions of non-commutative probability, namely free probability.Stemming from the notion of reduced free product [3,28] Voiculescu developed the theory of free probability in the 1980's with the goal of solving the free group factor problem. While this still remains unsolved, free probability has become a very important field of mathematical research. For further reading see [16,19]. This paper culminates in Theorem 3.2, that n freely independent contractions do indeed dilate to n freely independent unitaries. In a dilation theory context this has been done by Boca in [5] where he gives the most general unitary dilation of n contractions. The only free probability dilation result we know of is the unitary dilation of L-free sets of contractions of Popa and Vaes [22].Acknowledgements. We would like to thank David Sherman and Stuart White for some very helpful discussions during White's visit to the University of Virginia sponsored by the Institute of Mathematical Science.2010 Mathematics Subject Classification. 47A20, 46L54, 46L09.

Graph products of completely positive maps

“…where for each v ∈ V , x v denotes the unitary corresponding to the canonical generator of the v th copy of Z. Lastly, we have a version of Theorem 4.5 viewed through the lens of noncommutative probability. The statement and proof are simple adaptations of the free versions presented in [2]. Theorem 4.10.…”