Yes, that was a typo. It is "MP-partial isometry", short for "Moore-Penrose partial isometry".

@Bedovlat Sure, but you asked about positive linear functionals in $A^*$, and these are always bounded.

Note that a positive, linear functional on a unital C*-algebra $A$ is a trace if (and only if) it is invariant under all inner automorphisms. Thus, every $Aut(A)$-invariant functional is necessarily a trace. Moreover, by scaling, one may assume it is a tracial state. Let us consider the compact, convex set $T(A)$ of tracial states. The automorphism group naturally acts on it, and you are looking precisely for the fixed points of this action.

@DavidGao Thank you for picking this up (and for your nice answer to Question 1). I see your point that $\pi(b)$ may not agree with $b$. One only has $b \leq \pi(b)$. But it seems to me that $A$ is nevertheless monotone complete, with the supremum in $A$ given by $\pi(b)$. Does that sound OK to you?

Yes, this follows from Corollary 3 in [1]: Every compact hermitian operator is a sum of two self-commutators of compact operators. [1] Fan, Fong. Which operators are the self-commutators of compact operators? Proc. Amer. Math. Soc. 80 (1980), no. 1, 58–60