In this talk, I shall present the theory of Campanato spaces associated with quantum Markov semigroups on finite von Neumann algebras. One main result is that the column semigroup Campanato space is surprisingly completely isomorphic to the row one, which should have not been expected in noncommutative analysis, since the essential difference between the column space and the row space have been verified in many cases such as Hardy spaces and BMO spaces etc. The approach by identifying both of the spaces with semigroup Lipschitz space sheds new light on classical theory of harmonic analysis and function spaces. Also this identification happens to provide a positive answer to one open problem. Based on a joint work with Yuanyuan Jing.

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