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Why is every positive linear functional bounded in $C^*$-algebras?

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See page 11: – 1015 Feb 6 '13 at 21:27
In general, every positive $ \ast $-homomorphism from a C$ ^{\ast} $-algebra to another is necessarily bounded. – Haskell Curry Feb 6 '13 at 21:39
@Haskell: but functionals are not homomorphisms, so why "in general"? Also, note that every $*$-homomorphism is positive. – Martin Argerami Feb 16 '13 at 14:20
up vote 4 down vote accepted

For self-adjoint elements $a$, we have the inequality $-\lVert a\rVert e\le a\le\lVert a\rVert e$, where $e$ is the identity. So if $f$ is a positive linear functional, $-\lVert a\rVert f(e)\le f(a)\le\lVert a\rVert f(e)$ follows; i.e., $\lvert f(a)\rvert\le\lVert a\rVert f(e)$. For non-selfadjoint $a$, write $a=b+ic$ with $b$ and $c$ selfadjoint and use the result just shown.

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