A Generalization of Powers–Størmer Inequality

• Published in: Letters in mathematical physics Vol. 97; no. 3; pp. 339 - 346
• Main Author: Ogata, Yoshiko
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-09-2011
• Subjects: Complex Systems; Geometry; Group Theory and Generalizations; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Powers–Størmer inequality; Chernoff bound; 46L10
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-011-0504-y

In this note, we prove the following inequality:
, where
and
η
are positive normal linear functionals over a von Neumann algebra. This is a generalization of the famous Powers–Størmer inequality (Powers and Størmer proved the inequality for
in Commun Math Phys 16:1–33,
1970
; Takesaki in Theory of Operator Algebras II,
2001
). For matrices, this inequality was proven by Audenaert et al. (Phys Rev Lett 98:160501,
2007
). We extend their result to general von Neumann algebras.