Let $T_+$ be the set of a positive trace-class operators over some separable Hilbert space and

$A: T_+ to mathbb{R}cup {infty}$ some linear functional.

In general, $A$ will not be continuous. However, I’m interested in the following weak type of continuity of $A$: for any sequence ${P_n}$ of commuting and increasing projection operators ($P_n P_m = P_m P_n$, and $P_n ge P_m$ if $nge m$) that converge to the identity,

$$lim_{nto infty} A(P_n rho P_n) = A(rho) qquad forall rho in T_+.$$

For example, the von Neumann entropy $A(rho):=mathrm{tr}{-rho ln rho}$ is a functional (albeit a nonlinear one) that obeys this kind of continuity (Wehrl, 1978), although generally it is not continuous (only lower semi-continuous) in the trace-norm.

Is there a name for this weaker kind of continuity? Are there simple characterization of the class of functionals that obey it?