# The Modular Symmetry of Markov Maps

## TL;DR

This paper extends the understanding of the modular symmetry properties of state-preserving maps on von Neumann algebras, showing that unital completely positive maps also exhibit a canonical modular structure.

## Contribution

It generalizes known modular symmetry results from automorphisms to unital completely positive maps, revealing a broader modular framework.

## Key findings

- Unital completely positive maps admit a canonical modular structure.
- The modular symmetry extends beyond automorphisms to more general maps.
- The results unify the treatment of automorphisms and completely positive maps in modular theory.

## Abstract

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.