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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.
MSC:
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
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