Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1977-0442702-9
MathSciNet review: 0442702
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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).


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L05

Retrieve articles in all journals with MSC: 46L05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0442702-9
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