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Extensions, restrictions, and representations of states on $ C\sp{\ast} $-algebras


Author: Joel Anderson
Journal: Trans. Amer. Math. Soc. 249 (1979), 303-329
MSC: Primary 46L30
DOI: https://doi.org/10.1090/S0002-9947-1979-0525675-1
MathSciNet review: 525675
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Abstract: In the first three sections the question of when a pure state g on a $ {C^{\ast}}$-subalgebra B of a $ {C^{\ast}}$-algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf $ \left\Vert {b\left( {a\, - \,f\left( a \right)1} \right)b} \right\Vert\, = \,0$ for each a in A, where the inf is taken over those b in B such that $ 0\, \leqslant \,b\, \leqslant \,1$ and $ g(b) = 1$. The special cases where B is maximal abelian and/or $ A\, = \,B\left( H \right)$ are treated in more detail. In the remaining sections states of the form $ T \mapsto \mathop {\lim }\limits_{\mathcal{u}} \left( {T{x_\alpha },\,{x_\alpha }} \right)$, where $ \left\{ {{x_\alpha }} \right\}{\,_{\alpha \, \in \,\kappa }}$ is a set of unit vectors in H and $ {\mathcal{u}}$ is an ultrafilter are studied.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0525675-1
Keywords: Pure state, singular state, pure state extension, restriction, maximal abelian subalgebra, commutator, compression, conditional expectation, ultrafilter, Stone-Čech compactification, $ \beta {\text{N}}$, rare ultrafilter, p-point, selective ultrafilter
Article copyright: © Copyright 1979 American Mathematical Society

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