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The \(\lambda \)-Function in the Space of Trace Class Operators

  • Antonio M. Peralta
Article
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Abstract

Let \(C_1(H)\) denote the space of all trace class operators on an arbitrary complex Hilbert space H. We prove that \(C_1(H)\) satisfies the \(\lambda \)-property, and we determine the form of the \(\lambda \)-function of Aron and Lohman on the closed unit ball of \(C_1(H)\) by showing that
$$\begin{aligned} \lambda (a) = \frac{1 - \Vert a\Vert _1 + 2 \Vert a\Vert _{\infty }}{2}, \end{aligned}$$
for every a in \({C_1(H)}\) with \(\Vert a\Vert _1 \le 1\). This is a non-commutative extension of the formula established by Aron and Lohman for \(\ell _1\).

Mathematics Subject Classification

46B20 46E30 46L70 46L05 47A05 47C10 

Notes

Acknowledgements

The author is partially supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund project no. MTM2014-58984-P and Junta de Andalucía grant FQM375.

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Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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