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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Polar decompositions of bounded linear functionals on operator subalgebras


Author: Masaharu Kusuda
Journal: Proc. Amer. Math. Soc. 118 (1993), 839-843
MSC: Primary 46L10
MathSciNet review: 1149973
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Abstract: Let $ M$ be a von Neumann algebra and let $ \varphi $ be a normal linear functional on a strongly closed $ {C^{\ast}}$-subalgebra $ N$ of $ M$. Denote by $ {\mathcal{F}_\varphi }$ the set of normal linear functionals $ \psi $ on $ M$ extending $ \varphi $ with $ \vert\vert\psi \vert\vert = \vert\vert\varphi \vert\vert$. It is shown that there exists a partial isometry $ v$ in $ N$ such that

$\displaystyle \varphi = \vert\varphi \vert(v \cdot ),\qquad \vert\varphi \vert ... ... ),\qquad \vert\vert\varphi \vert\vert = \vert\vert\vert\varphi \vert\vert\vert$

and

$\displaystyle \psi = \vert\psi \vert(v\cdot),\qquad \vert\psi \vert = \psi ({v^... ...}}\cdot),\qquad \vert\vert\psi \vert\vert = \vert\vert\vert\psi \vert\vert\vert$

for all $ \psi $ in $ {\mathcal{F}_\varphi }$, where $ \vert\varphi \vert$ and $ \vert\psi \vert$ denote the absolute values of $ \varphi $ and $ \psi $ respectively. Let $ A$ be a $ {C^{\ast}}$-algebra and let $ B$ be a $ {C^{\ast}}$-subalgebra of $ A$. As a consequence of this result, we obtain that every state on $ B$ has a unique state extension to $ A$ if and only if every bounded linear functional on $ B$ has a unique norm-preserving extension to a bounded linear functional on $ A$.

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DOI: https://doi.org/10.1090/S0002-9939-1993-1149973-9
Article copyright: © Copyright 1993 American Mathematical Society