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 $$\lambda (a)=\frac{1-\mathrm{\parallel }a{\mathrm{\parallel }}_1+2\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}}{2},$$ for every $a$ in $C_1(H)$ with $\mathrm{\parallel }a{\mathrm{\parallel }}_1\le 1$. This is a non-commutative extension of the formula established by Aron and Lohman for ${\mathrm{\ell }}_1$.