Automatic continuity of M-norms on C*-algebras

Elements a and b of a C*-algebra are called orthogonal (a perpendicular to b) if a*b = ab* = 0. We say that vectors x and y in a Banach space X are semi-M-orthogonal (x perpendicular to (SM) y) if parallel to x +/- y parallel to >= max{parallel to x parallel to, parallel to y parallel to}. 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) perpendicular to(SM) T(b) whenever a perpendicular to b, must be continuous. Consequently, every complete (semi-)M-norm on a von Neumann algebra or on a compact C*-algebra is automatically continuous. (C) 2011 Elsevier Inc. All rights reserved.