Brown–Voiculescu entropy revisited

The paper proves Theorem 5.4 rigorously: for simple, separable, unital, nuclear, Z-stable C*-algebras with finitely many extremal traces and UCT, not AF, one has htdr(α) = sup_{τ∈∂eT(A)} hqd_τ(α) = sup_{τ∈T(A)} hqd_τ(α). The proof first shows htdr ≥ hqd_τ for every trace (Proposition 5.3), then constructs 2-colour, contractive decomposable approximations from quasidiagonal systems of the finitely many extremal traces, yielding htdr ≤ max_i hqd_{τ_i} and hence the equality (Theorem 5.4, with the key steps (5.1)–(5.7)) . By contrast, the model solution conflates nuclear dimension with decomposition rank in step (i), incorrectly claiming CETWW’s dim_nuc(A)=1 gives contractive two-colour (decomposition rank) approximations; contractivity is part of decomposition rank, not nuclear dimension. It also incorrectly asserts that in the AF case both sides are zero, contradicting the paper’s AF computations where htdr equals Voiculescu’s approximation entropy and can be positive (e.g., log k for a UHF Bernoulli shift) .