Automatic continuity of orthogonality preservers on a non-commutative L~p(τ) space

摘要

Elements a and b of a non-commutative L~p(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 L~p(N, τ) to a normed space Y is said to be orthogonality-to-p-orthogonality preserving if {norm of matrix}T(a)+T(b){norm of matrix}~p={norm of matrix}a{norm of matrix}~p+{norm of matrix}b{norm of matrix}~p whenever a and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from L~p(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 I_2, 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 {norm of matrix}T{norm of matrix}{norm of matrix}T-1{norm of matrix}≤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.