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

Let $\mathcal{A}$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra). 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 $\|a\|=\sqrt{R_{\sigma}(a^{*}a)}$.

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 NormAndSpectralRadiusInCalgebras 2013-03-22 17:38:41 2013-03-22 17:38:41 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 46L05 HomomorphismsOfCAlgebrasAreContinuous CAlgebra