## 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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