Norm inequalities in star algebras. (English) Zbl 1014.46031

The author proves that a unital \(W^*\)-algebra \(X\) of operators is noncommutative if the inequality \[ \|A+B\|>1+\|AB\| \] holds for all self-adjoint elements \(A,B\) of \(X\) with \(\|A\|= \|B \|=1\). Making use of the proof of the above theorem, he proves that for \(\lambda\), \(\sigma >0\) and \(\mu\in \mathbb{R}\) there exist elements \(T_1,T_2,T_3\) of a given \(C^*\)-algebra \(X\) of operators such that \(\lambda T_1+\mu T_2+ \sigma T_3\geq 0\) if and only if \(\lambda\sigma \geq\mu^2\), and this result is related to the Furuta inequality. Furthermore, he gives some examples as an application of these inequalities in the case of \(C^*\)-algebras.


46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
Full Text: EuDML