$C^{*}$-algebra homomorphisms are continuous

Theorem - Let $\mathcal{A},\mathcal{B}$ be $C^{*}$-algebras (http://planetmath.org/CAlgebra) and $f:\mathcal{A}\longrightarrow\mathcal{B}$ a *-homomorphism. Then $f$ is bounded (http://planetmath.org/ContinuousLinearMapping) and $\|f\|\leq 1$ (where $\|f\|$ is the norm (http://planetmath.org/OperatorNorm) of $f$ seen as a linear operator between the spaces $\mathcal{A}$ and $\mathcal{B}$).

For this reason it is often said that homomorphisms between $C^{*}$-algebras are automatically continuous (http://planetmath.org/ContinuousLinearMapping).

Corollary - A *-isomorphism between $C^{*}$-algebras is an isometric isomorphism (http://planetmath.org/IsometricIsomorphism).
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Proof of Theorem : Let us first suppose that $\mathcal{A}$ and $\mathcal{B}$ have identity elements, both denoted by $e$.

We denote by $\sigma(x)$ and $R_{\sigma}(x)$ the spectrum and the spectral radius of an element $x\in\mathcal{A}$ or $\mathcal{B}$.

Let $a\in\mathcal{A}$ and $\lambda\in\mathbb{C}$. If $a-\lambda e$ is invertible in $\mathcal{A}$, then $f(a-\lambda e)$ is invertible in $\mathcal{B}$. Thus,

 $\sigma(f(a))\subseteq\sigma(a)\,.$

Hence $R_{\sigma}(f(a))\leq R_{\sigma}(a)$ for every $a\in\mathcal{A}$. Therefore, by the result from this entry (http://planetmath.org/NormAndSpectralRadiusInCAlgebras),

 $\|f(a)\|=\sqrt{R_{\sigma}(f(a)^{*}f(a))}=\sqrt{R_{\sigma}(f(a^{*}a))}\leq\sqrt% {R_{\sigma}(a^{*}a)}=\|a\|\,.$

We conclude that $f$ is and $\|f\|\leq 1$.

If $\mathcal{A}$ or $\mathcal{B}$ do not have identity elements, we can consider their minimal unitizations, and the result follows from the above . $\square$

Proof of Corollary : This follows from the fact that $f^{-1}$ is also a *-homomorphism and therefore $\|f^{-1}(b)\|\leq\|b\|$ for every $b\in\mathcal{B}$. $\square$

Title $C^{*}$-algebra homomorphisms are continuous CalgebraHomomorphismsAreContinuous 2013-03-22 17:40:06 2013-03-22 17:40:06 asteroid (17536) asteroid (17536) 14 asteroid (17536) Theorem msc 81R15 msc 46L05 automatic continuity of $C^{*}$-homomorphisms homomorphisms of $C^{*}$-algebras are continuous ContinuousLinearMapping OperatorNorm C_cG UniformContinuityOverLocallyCompactQuantumGroupoids CAlgebra CAlgebra3 NormAndSpectralRadiusInCAlgebras EquivalenceOfDefinitionsOfCAlgebra GroupoidCConvolutionAlgebra automatically continuous homomorphism of $C^{*}$–algebras