[Orthogonalisation de partitions de l’unité]
Nous montrons qu’une partition de l’unité dans un espace de Hilbert qui est presque orthogonale est proche d’une partition de l’unité orthogonale dans la même algèbre de von Neumann. Ce résultat affine et généralise à la dimension infinie des résultats antérieurs de Kempe–Vidick et Ji–Natarajan–Vidick–Wright–Yuen dans les algèbres de matrices. Quantitativement, nos résultats sont également plus fins puisque nous obtenons une dépendance linéaire, qui est optimale.
Nous généralisons également à la dimension infinie un autre résultat de dualité entre partitions de l’unité et majorants minimaux de parties finies dans le prédual d’une algèbre de von Neumann.
We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe–Vidick and Ji–Natarajan–Vidick–Wright–Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal.
We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.
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@article{CRMATH_2022__360_G5_549_0,
author = {de la Salle, Mikael},
title = {Orthogonalization of {Positive} {Operator} {Valued} {Measures}},
journal = {Comptes Rendus. Math\'ematique},
pages = {549--560},
publisher = {Acad\'emie des sciences, Paris},
volume = {360},
number = {G5},
year = {2022},
doi = {10.5802/crmath.326},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.326/}
}
TY - JOUR AU - de la Salle, Mikael TI - Orthogonalization of Positive Operator Valued Measures JO - Comptes Rendus. Mathématique PY - 2022 SP - 549 EP - 560 VL - 360 IS - G5 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.326/ DO - 10.5802/crmath.326 LA - en ID - CRMATH_2022__360_G5_549_0 ER -
%0 Journal Article %A de la Salle, Mikael %T Orthogonalization of Positive Operator Valued Measures %J Comptes Rendus. Mathématique %D 2022 %P 549-560 %V 360 %N G5 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.326/ %R 10.5802/crmath.326 %G en %F CRMATH_2022__360_G5_549_0
de la Salle, Mikael. Orthogonalization of Positive Operator Valued Measures. Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 549-560. doi : 10.5802/crmath.326. http://www.numdam.org/articles/10.5802/crmath.326/
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