Automatic continuity of orthogonality preservers on a non-commutative Lp(τ ) space

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. © 2013 Elsevier Inc. All rights reserved