Theoretical Mathematics & Applications

E. Lieb convexity inequalities and noncommutative Bernstein inequality in Jordan-algebraic setting

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• Abstract

  We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. A version of noncommutative Bernstein inequality is proven as an application of one of convexity inequalities. A spectral theory on semi-simple complex Jordan algebras is used as a tool to prove the convexity results. Possible applications to optimization and statistics are indicated.