Isometries of spaces of normalized positive operators under the operator norm

In this paper, a former result of ours [13, Theorem 2] is completed. It asserts that for all real numbers $p>1$, the $p$-norm isometries of the space of elements with $p$-norm 1 in the cone of positive operators on a finite dimensional complex Hilbert space are unitary or antiunitary conjugations. The purpose of this paper is to provide an analogous statement in the case $p=\infty$, i.e., the case of the operator norm.