# Graph products of completely positive maps

## TL;DR

This paper introduces the graph product of unital completely positive maps on C*-algebras, proving it remains completely positive and exploring its implications for positive-definite functions, inequalities, and dilations.

## Contribution

It defines the graph product of completely positive maps, proves its complete positivity, and establishes new results on positive-definite functions and dilations in this context.

## Key findings

- Graph product of positive-definite functions is positive-definite
- A graph product version of von Neumann's Inequality holds
- Graph independent contractions dilate to graph independent unitaries

## Abstract

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.