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We extend Voiculescu's approach to analytic subordination through thecoalgebra of the free difference quotient to non-coassociativederivation-comultiplications appearing in free probability theory. We obtainnew proofs of Voiculescu's analytic subordination results for freely Markoviantriples, and for multiplication of unitaries which are free with amalgamation.

Analytic subordination results in free probability from non-coassociative derivation-comultiplications