# reference request – Continuity of a linear functional for sequences of projections

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?