Counterexamples to the extendibility of positive unital norm-one maps - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Linear Algebra Appl. Year : 2023

Counterexamples to the extendibility of positive unital norm-one maps

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.

Keywords

Dates and versions

hal-03659614 , version 1 (05-05-2022)

Identifiers

Cite

Giulio Chiribella, Kenneth R. Davidson, Vern I. Paulsen, Mizanur Rahaman. Counterexamples to the extendibility of positive unital norm-one maps. Linear Algebra Appl., 2023, 663, pp.102-115. ⟨10.1016/j.laa.2023.01.003⟩. ⟨hal-03659614⟩
33 View
0 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More