Article

# Operators preserving orthogonality are isometries

Proceedings of the Royal Society of Edinburgh Section A Mathematics (Impact Factor: 0.64). 01/1993; DOI: 10.1017/S0308210500029528

Source: arXiv

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**ABSTRACT:**Elements a and b of a C⁎C⁎-algebra are called orthogonal (a⊥ba⊥b) if a⁎b=ab⁎=0a⁎b=ab⁎=0. We say that vectors x and y in a Banach space X are semi-M-orthogonal (xSM⊥yx⊥SMy) if ‖x±y‖⩾max{‖x‖,‖y‖}‖x±y‖⩾max{‖x‖,‖y‖}. We prove that every linear bijection T:A→XT:A→X, where X is a Banach space, A is either a von Neumann algebra or a compact C⁎C⁎-algebra, and T(a)SM⊥T(b)T(a)⊥SMT(b) whenever a⊥ba⊥b, must be continuous. Consequently, every complete (semi-)M-norm on a von Neumann algebra or on a compact C⁎C⁎-algebra is automatically continuous.Journal of Mathematical Analysis and Applications 09/2011; 381(2):799–811. · 1.12 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We define a ρ-orthogonality in a real normed space and we consider the class of linear mappings preserving this relation. We show that a linear mapping preserving ρ-orthogonality has to be a similarity, i.e., a scalar multiple of an isometry. As a result, we give a characterization of smooth spaces in terms of this orthogonality.Journal of Mathematical Analysis and Applications 02/2012; 386(1):171–176. · 1.12 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We survey mainly recent results on the two most important orthogonality types in normed linear spaces, namely on Birkhoff orthogonality and on isosceles (or James) orthogonality. We lay special emphasis on their fundamental properties, on their differences and connections, and on geometric results and problems inspired by the respective theoretical framework. At the beginning we also present other interesting types of orthogonality. This survey can also be taken as an update of existing related representations.Aequationes Mathematicae 01/2012; 83(1-2). · 0.55 Impact Factor

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