How did Lang deduce norm preserving here?

Question

This is related to Lang's Real and Functional Analysis Chpt XVI, sec 3, Thm 3.3 proof.

Thm 3.3. $A$ is commutative $C^\star$ algebra with unit. $A\to C(\hat{A},C)$ given by evaluation map is norm preserving.

Consider $x\in A$ and $y=x^\star x$. Then $y^\star=y$ where $\star$ is involution of $A$. It follows from following Prop 3.1 that $|y|=\rho(y)$ where $\rho(y)$ is spectral radius of $y$.

Prop 3.1 $A$ is $C^\star$ algebra with unit. $x\in A$ and if $x=x^\star$, then spectrum of $x$ is real and $\rho(x)=|x|$.

Thm 1.7 $A$ is banach algebra with non-zero unit. Then $\forall x\in A$, $\rho(x)=\sup_{z\in Sp(x)}|z|$ where $Sp(x)=\{z\in C\vert x-z\cdot 1$ not invertible$\}$.

$\textbf{Q:}$ The book says by Thm 1.7 and Prop 3.1, we find $|g(y)|=\rho(y)=|y|$ where $g(y)$ denotes an evaluation of $y$. Why $|g(y)|=\rho(y)$? Suppose $g\in\hat{A}$, then $g(y)\in Sp(y)$ for sure. Hence $g(y)\leq\rho(y)$. Why do I have a sequence of $g\in\hat{A}$ s.t. $|g(y)|=\rho(y)$?

Answer 1

Your theorem is the following:

Let $A$ be a commutative $C^*$ algebra and consider the map $F:A\to C(\hat A,\Bbb C)$, $y\mapsto (\varphi\mapsto \varphi(y))$. Then $F$ preserves the norm.

The necessary statements to prove this are:

1. For any $r\in\mathrm{Sp}(y)$ there is a character $\varphi\in\hat A$ with $\varphi(y)=r$.

2. If $y\in A$ and $y^*=y$ then $\|y\|=\sup_{r\in\mathrm{Sp}(y)}|r|$.

3. If $x\in A$ then $\|x\|^2=\|x^*x\|$.

Note that you have already written down 2. and 3.

We now use these statements. If $x\in A$, note that: $$\|F(x)\|=\sup_{\varphi\in\hat A}|\varphi(x)| = \sqrt{\sup_{\varphi\in \hat A}|\varphi(x)|^2}=\sqrt{\sup_{\varphi\in \hat A}|\varphi(x^*x)|}$$ Now by 1. for any $r\in\mathrm{Sp}(x^*x)$ there exists a $\varphi\in \hat A$ so that $\varphi(x^*x)=r$, hence by 2. you have that the supremum on the right is $≥\sup_{r\in\mathrm{Sp}(x^*x) }|r|=\|x^*x\|=\|x\|^2$, taking the root then gives: $$\|F(x)\|=\sup_{\varphi\in\hat A}|\varphi(x)|≥\|x\|.$$

The remaining inequality $$\|F(x)\|≤\|x\|$$ follows by a general property of $C^*$-morphisms:

4. If $F:A\to B$ is a $*$-morphism between two $C^*$-algebras $A$ and $B$, then $\|F(x)\|_B≤\|x\|_A$ for all $x\in A$.