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[parent] norm and spectral radius in $C^*$-algebras (Theorem)

Let $ \mathcal{A}$ be a $ C^*$-algebra. Let $ R_{\sigma}(a)$ denote the spectral radius of the element $ a \in \mathcal{A}$.

Theorem - For every $ a \in \mathcal{A}$ we have that $ \Vert a\Vert = \sqrt{R_{\sigma}(a^*a)}$.

This result shows that the norm in a $ C^*$-algebra has a purely algebraic nature. Moreover, the norm in a $ C^*$-algebra is unique (in the sense that there is no other norm for which the algebra is a $ C^*$-algebra).



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See Also: $C^*$-algebra homomorphisms are continuous, $C^*$-algebra


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Cross-references: algebra, norm, spectral radius
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This is version 4 of norm and spectral radius in $C^*$-algebras, born on 2007-11-29, modified 2007-11-30.
Object id is 10067, canonical name is NormAndSpectralRadiusInCAlgebras.
Accessed 395 times total.

Classification:
AMS MSC: 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
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