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

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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 \(\ell _1\).

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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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Correspondence to Antonio M. Peralta.

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Peralta, A.M. The \(\lambda \)-Function in the Space of Trace Class Operators. Mediterr. J. Math. 15, 217 (2018). https://doi.org/10.1007/s00009-018-1260-3

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