Counterexamples to the extendibility of positive unital norm-one maps

• URL: http://arxiv.org/abs/2204.08819v1
• Date: Tue, 19 Apr 2022 11:40:41 GMT
• Title: Counterexamples to the extendibility of positive unital norm-one maps
• Authors: Giulio Chiribella, Kenneth R. Davidson, Vern I. Paulsen and Mizanur Rahaman
• Abstract summary: 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. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem cannot be extended to a positive map on the full matrix algebra.
• Score: 5.926203312586108
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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