Build a C*-Algebra from the set of Continuous Functions

Question

Let $X$ be a compact Hausdorff space.

Let $A$ be a $C^*$-algebra.

Consider $C(X,A) = \{ f: X \rightarrow A : f \text{ is continuous }\}$.

Suppose $f^*(x) = (f(x))^*$ and $\|f\| = \sup \{ \|f(x)\| : x \in X\}$.

Show that $C(X,A)$ is a $C^*$-algebra.

I know that if we use pointwise sum and product, then $C(X,A)$ is an algebra.

Now I have to show that if $f\in C(X,A)$, then $\| f^* f\| = \|f\|^2$.

We have $\| f^* f\| = \sup\{ \| f^* f (x)\| \} \leq \sup\{ \| f^*(x)\| \|f (x)\| \} = \| f \|^2$.

Also, $\| f\|^2 = \sup \{ \| f(x)\|^2 \}$...

But I'm not sure how to proceed from here. Thank you in advance!

Answer 1

You have $$ \|f^*f\|=\sup\{\|f(x)^*f(x)\|:\ x\in X\}=\sup\{\|f(x)\|^2:\ x\in X\}=\big[\sup\{\|f(x)\|:\ x\in X\}\big]^2=\|f\|^2. $$