Counterexamples to the extendibility of positive unital norm-one maps
| Autor: | Chiribella, Giulio, Davidson, Kenneth R., Paulsen, Vern I., Rahaman, Mizanur |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Linear Algebra and its Applications (2023) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.laa.2023.01.003 |
| Popis: | 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. Comment: Comments are welcome |
| Databáze: | arXiv |
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