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

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 \(\ell _1\).