Automatic continuity of orthogonality preservers on a non-commutative [formula omitted] space

Abstract

Elements a and b of a non-commutative Lp(τ) space associated to a von Neumann algebra N, equipped with a normal semifinite 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(N,τ) to a normed space Y 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(N,τ) (1⩽p<∞, p≠2) to a Banach space X is automatically continuous, whenever N is a separably acting von Neumann algebra. If N is a semifinite factor not of type I2, we establish that every orthogonality-to-p-orthogonality preserving linear mapping T:Lp(N,τ)→X is continuous, and invertible whenever T≠0. Furthermore, there exists a positive constant C(p) (1⩽p<∞, p≠2) so that ‖T‖‖T−1‖⩽C(p)2, for every non-zero orthogonality-to-p-orthogonality preserving linear mapping T:Lp(N,τ)→X. For p=1, this inequality holds with C(p)=1 – that is, T is a multiple of an isometry.