Automatic continuity of M-norms on C⁎-algebras

Elements a and b of a C⁎-algebra are called orthogonal (a⊥b) if a⁎b=ab⁎=0. We say that vectors x and y in a Banach space X are semi-M-orthogonal (x⊥SMy) 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)⊥SMT(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.