Positive linear maps that converge on a small test set of continuous functions converge on the whole space, even when the limit is a nonlinear isometry.
Donner, Extension of Positive Operators and Korovkin Theorems , Lecture Notes in Math
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Korovkin-type results on convergence of sequences of positive linear maps on function spaces
Positive linear maps that converge on a small test set of continuous functions converge on the whole space, even when the limit is a nonlinear isometry.