# Journal of Operator Theory

Volume 81, Issue 1, Winter 2019  pp. 133-156.

Graph products of completely positive maps

Authors:  Scott Atkinson
Author institution: Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, U.S.A.

Summary:  We define the graph product of unital completely positive maps on a universal graph product of unital $C^*$-algebras and show that it is unital completely positive itself. To accomplish this, we introduce the notion of the non-commutative length of a word, and we obtain a Stinespring construction for concatenation. This result yields the following consequences. The graph product of positive-definite functions is positive-definite. A graph product version of von Neumann's inequality holds. Graph independent contractions on a Hilbert space simultaneously dilate to graph independent unitaries.

DOI: http://dx.doi.org/10.7900/jot.2017dec13.2177
Keywords:  completely positive maps, graph products, $C^*$-algebras