Automatic continuity of M-norms on -algebras

Elements a and b of a C ⁎ -algebra are called orthogonal ( a ⊥ b ) if a ⁎ b = a b ⁎ = 0 . We say that vectors x and y in a Banach space X are semi-M-orthogonal ( x ⊥ SM y ) if ‖ x ± y ‖ ⩾ max { ‖ x ‖ , ‖ y ‖ } . We prove that every linear bijection T : A → X , where X is a Banach space, A is either a von Neumann algebra or a compact C ⁎ -algebra, and T ( a ) ⊥ SM T ( b ) whenever a ⊥ b , must be continuous. Consequently, every complete (semi-)M-norm on a von Neumann algebra or on a compact C ⁎ -algebra is automatically continuous.