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Trace inequalities for Rickart $$C^*$$ C ∗ -algebras
Positivity ( IF 1 ) Pub Date : 2021-07-24 , DOI: 10.1007/s11117-021-00852-3
Airat Bikchentaev 1
Affiliation  

Rickart \(C^*\)-algebras are unital and satisfy polar decomposition. We proved that if a unital \(C^*\)-algebra \({\mathcal {A}}\) satisfies polar decomposition and admits “good” faithful tracial states then \({\mathcal {A}}\) is a Rickart \(C^*\)-algebra. Via polar decomposition we characterized tracial states among all states on a Rickart \(C^*\)-algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart \(C^*\)-algebra. For a block projection operator and a trace on a Rickart \(C^*\)-algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra.



中文翻译:

跟踪 Rickart 的不等式 $$C^*$$ C ∗ -代数

Rickart \(C^*\) -代数是单位的并且满足极性分解。我们证明了如果单位\(C^*\) -代数\({\mathcal {A}}\)满足极分解并承认“好”忠实轨迹状态,那么\({\mathcal {A}}\)是一个 Rickart \(C^*\) -代数。通过极性分解,我们在 Rickart \(C^*\) -代数上表征了所有状态中的轨迹状态。我们在 Rickart \(C^*\) -代数上提出了 Hermitian 元素和迹的三角不等式。对于块投影算子和 Rickart 上的迹线\(C^*\)-代数我们证明了一个新的不等式。作为推论,我们获得了对任何厄米元素的换向器迹线和投影的精确估计。此外,我们还给出了冯诺依曼代数上各种权重的迹的表征。

更新日期:2021-07-24
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