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Proof Linear Transformation Types Of Homomorphisms Identity Element Something Related To Injective Function Operator Norm Continuous Linear Mapping Spectral Radius C Algebra Category Of C Algebras Isometric Isomorphism Norm And Spectral Radius In C Algebras Unitization

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Homomorphisms Of C Algebras Are Continuous

Theorem - Let $\mathcal{A}, \mathcal{B}$ be -algebras and $f:\mathcal{A} \longrightarrow \mathcal{B}$ a *-homomorphism. Then is bounded and $\Vert f\Vert \leq 1$ (where $\Vert f\Vert$ is the norm of seen as a linear operator between the spaces $\mathcal{A}$ and $\mathcal{B}$ ).

For this reason it is often said that homomorphisms between -algebras are automatically continuous.

Corollary - A *-isomorphism between -algebras is an isometric isomorphism.
$\;$

Proof of Theorem : Let us first suppose that $\mathcal{A}$ and $\mathcal{B}$ have identity elements, both denoted by .

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,

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

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

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

We conclude that is bounded and $\Vert f\Vert \leq 1$ .

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

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

1	Types Of Homomorphisms
1	Linear Transformation
1	Proof
1	Something Related To Injective Function
1	Identity Element
4	Spectral Radius
4	Operator Norm
4	Continuous Linear Mapping
5	C Algebra
8	Category Of C Algebras
8	Isometric Isomorphism
13	Unitization
18	Norm And Spectral Radius In C Algebras