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Continuing the analysis in [19, 5] , we give purely algebraic characterization of the canonical endomorphism in interesting infinite index cases. We apply these results when compact and discrete (but not necessarily finite dimensional) Woronowicz algebras  act alternately on the factors in the various levels of Jones' tower. We characterise when the acting algebra is a Kac algebra. Mathematics Subject Classification Numbers: Primary 46L37, Secondary 46L05.doi:10.4171/zaa/869 fatcat:y4ymf6mdavg77kntkasocnzdku