Automatic continuity of orthogonality or disjointness preserving bijections

Elements a and b of a non-commutative Lp(M,τ) space associated to a von Neumann algebra, M, equipped with a normal semi-finite faithful trace τ, are called orthogonal if l(a)l(b)=r(a)r(b)=0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from Lp(M,τ) to a normed space X is said to be orthogonality-to-p-orthogonality preserving if ∥T(a)+T(b)∥p=∥a∥p+∥b∥p whenever a and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(M,τ) to a Banach space is automatically continuous if 1≤p