Counterexamples to the extendibility of positive unital norm-one maps

TL;DR

This paper presents three counterexamples demonstrating that positive unital norm-one maps on operator subsystems of matrix algebras cannot always be extended to the entire algebra, highlighting limitations of extendibility beyond complete positivity.

Contribution

The paper provides the first explicit counterexamples showing that positivity and unital norm-one conditions do not guarantee extendibility to full matrix algebras.

Findings

Counterexample of a positive unital map with unit norm that cannot be extended

Counterexample of a positive unital isometry on a real operator space that cannot be extended

Counterexample of a positive unital isometry on a complex operator space that cannot be extended

Abstract

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.