# 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

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