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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Decompositions of linear maps

Author: Sze Kai J. Tsui
Journal: Trans. Amer. Math. Soc. 230 (1977), 87-112
MSC: Primary 46L05
MathSciNet review: 0442702
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Abstract: In the first part we show that the decomposition of a bounded selfadjoint linear map from a $ {C^\ast}$-algebra into a given von Neumann algebra as a difference of two bounded positive linear maps is always possible if and only if that range algebra is a ``strictly finite'' von Neumann algebra of type I. In the second part we define a ``polar decomposition'' for some bounded linear maps and show that polar decomposition is possible if and only if the map satisfies a certain ``norm condition". We combine the concepts of polar and positive decompositions to show that polar decomposition for a selfadjoint map is equivalent to a strict Hahn-Jordan decomposition (see Theorems 2.2.4 and 2.2.8).

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Keywords: Positive linear maps, completely positive linear maps, positive decomposition, projective tensor product, Clifford algebra, Stonean spaces, type $ {\text{II}_1}$ von Neumann algebras, infinite von Neumann algebras, polar decomposition of linear maps, injective $ {C^\ast}$-algebras, partial isometries, norm-condition, Hahn-Jordan decomposition
Article copyright: © Copyright 1977 American Mathematical Society