Quantization of Branched Coverings Revisited

We address the problem of identification of branched coverings (continuous open surjections $$p\colon Y\to X$$ of Hausdorff spaces with uniformly bounded number of pre-images) with faithful unital positive conditional expectations $$E\colon C(Y)\to C(X)$$ topologically of index-finite type. Caused by recent progress (A. Chirvasitu, 2024) in the field and detection of an issue in our old paper, we revisit it to make some advances.