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W2007147823 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." @default.
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W2007147823 date "2013-04-01" @default.
W2007147823 modified "2023-10-16" @default.
W2007147823 title "Automatic continuity of orthogonality preservers on a non-commutative<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>space" @default.
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W2007147823 doi "https://doi.org/10.1016/j.jfa.2013.01.019" @default.
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