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Boundedness and compactness properties of multiplication operators on quantum (non-commutative) function spaces are investigated. For endomorphic multiplication operators these properties can be characterized in the setting of quantum symmetric spaces. For non-endomorphic multiplication operators these properties can be completely characterized in the setting of quantum L pspaces and a partial solution obtained in the more general setting of quantum Orlicz spaces.

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In this paper we characterize surjective isometries on certain classes of non-commutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces L w,1 , as well as the spaces L 1 + L ∞ and L 1 ∩ L ∞ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces.