# The $$\lambda$$-Function in the Space of Trace Class Operators

• Antonio M. Peralta
Article

## 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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