# Roots of Completely Positive Maps

## TL;DR

This paper explores the concept of roots of completely positive maps in operator algebras, providing structural results, examples, and open problems related to their existence and properties.

## Contribution

It introduces the notion of roots of completely positive maps in various forms and connects these ideas to classical and quantum probability problems.

## Key findings

- Structural and existence results for roots of completely positive maps
- Examples illustrating different types of roots
- Identification of open problems in the area

## Abstract

We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous one-parameter semigroups of roots. We present structural and general existence and non-existence results, some special examples in settings where we understand the situation better, and several challenging open problems. Our study is closely related to Elfving's embedding problem in classical probability and the divisibility problem of quantum channels.