Peripherally automorphic unital completely positive maps

TL;DR

This paper characterizes a special class of unital completely positive maps on finite-dimensional $C^*$-algebras, focusing on their spectral properties and algebraic structure, revealing conditions under which they are *-automorphisms.

Contribution

It provides a novel characterization of peripheral automorphic UCP maps and decomposes general UCP maps into persistent and transient components.

Findings

UCP maps with spectrum on the unit circle are *-automorphisms.

The Choi-Effros product extension matches the original product for certain UCP maps.

Decomposition into persistent and transient parts elucidates the structure of UCP maps.

Abstract

We identify and characterize unital completely positive (UCP) maps on finite dimensional $C^*$-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We analyze a decomposition of general UCP maps in finite dimensions into persistent and transient parts. It is shown that UCP maps on finite dimensional $C^*$-algebras with spectrum contained in the unit circle are $\ast$-automorphisms.