norm and spectral radius in $C^{*}$-algebras

Let $𝒜$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra). Let $R_{\sigma }(a)$ denote the spectral radius of the element $a\in 𝒜$.

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

This result shows that the norm in a $C^{*}$-algebra has a purely 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).

Title	norm and spectral radius in $C^{*}$-algebras
Canonical name	NormAndSpectralRadiusInCalgebras
Date of creation	2013-03-22 17:38:41
Last modified on	2013-03-22 17:38:41
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	7
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46L05
Related topic	HomomorphismsOfCAlgebrasAreContinuous
Related topic	CAlgebra