The norm of the difference of normal linear functionals

Question

Let $M$ be a von Neumann algebra. Suppose that $\varphi_1, \varphi_2\in M_{*}^{+}$, then we have

$||\varphi_1–\varphi_2||=\text{inf}\{2\rho(1)–\varphi_1(1)–\varphi_2(1)：\rho\geq \varphi_1,\rho\geq \varphi_2 \}$.

The above conclusion is from Haagerup and Stormer's paper. I wonder whether the infimum in the above relation can be attained.

In Haagerup and Stormer's paper, they mention that the infimumu can be attained, I cannot prove the conclusion.

Answer 1

In the lemma where they prove the equality, they explicitly say that the infimum is attained when $\rho=\varphi_1\vee \varphi_2=\varphi_2+(\varphi_1-\varphi_2)^+$.