Decompositions of linear maps

Tsui, Sze Kai J.

doi:10.1090/S0002-9947-1977-0442702-9

Decompositions of linear maps
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by Sze Kai J. Tsui PDF

Trans. Amer. Math. Soc. 230 (1977), 87-112 Request permission

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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