‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Question

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

‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎

‎let $‎‎C^*({a}) $ be the ‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎.

I know that ‎$‎‎C^*({a}) $ ‎is ‎the ‎closed ‎linear ‎span ‎of ‎‎$‎‎a^{m}a^{*{n}}$‎‎‎ such that $m,n\in N$.

‎ ‎ I want to know ‎$‎1 , a^{-1} \in ‎‎C^*({a}) ‎‎$‎‎ ‎

Q: Is it true?"$‎1 , a^{-1} \in ‎‎C^*({a}) ‎‎$‎‎"‎

How can I prove it? ‎‎

Answer 1

The spectrum of $a$ is a compact set that does not contain $0$. So there is a disk $D$ around $0$ with $D\cap\sigma(a)=\emptyset$. Thus, on $\sigma(a)$, $f:t\longmapsto 1/t$ is continuous, so $f\in C(\sigma(a))$. Then $f(a)\in C^*(a)$ via the Gelfand transform.

Edit: in view of Josse's comments, here's a clarification. Since $ a $ is invertible, $\sigma (a)\cap\{0\}=\varnothing $, so there exists a continuous function $f $ with $f (0)=0$ and $f (t)=1/t $ on $\sigma (a) $. By using Stone-Weierstrass on a closed disk, we can write $f $ as a uniform limit of polynomials (on $z $ and $\bar z $) with constant term zero. This implies that $a^{-1}\in C^*(a) $, and a fortiori $1\in C^*(a) $.

Answer 2

Let $B \subseteq A$ denote the $C^*$-subalgebra generated by $1$ and $a$. Recall that the functional calculus gives us a unital $*$-homomorphism $\varphi : C(\sigma(a)) \to A$ with the following properties:

• $\varphi(\iota) = a$, where $\iota : \sigma(a) \to \mathbb{C}$ denotes the inclusion $z \mapsto z$.
• $\varphi$ is isometric;
• The image of $\varphi$ is the $C^*$-subalgebra $B \subseteq A$ generated by $1$ and $a$.
• Consequently, $\varphi$ restricts to a $*$-isomorphism $C(\sigma(a)) \to B$.

Now, since we have $0\notin \sigma(a)$, we see that $\iota \in C(\sigma(a))$ vanishes nowhere and separates points:

• For all $x\in \sigma(a)$ we have $\iota(x) = x \neq 0$ (here we use that $0 \notin \sigma(a)$ holds);
• For all $x,y\in\sigma(a)$ with $x \neq y$ we have $\iota(x) \neq \iota(y)$.

As such, it follows from the Stone–Weierstrass theorem (locally compact version) that $\iota$ generates $C(\sigma(a))$ as a $C^*$-algebra. Since $\varphi$ restricts to a $*$-isomorphism $C(\sigma(a)) \to B$ with $\iota \mapsto a$, it follows that $B$ is generated by $a$. In other words, we have $B = C^*(a)$. Now it is immediate that $1 \in C^*(a) = B$ holds. Furthermore, it is now easy to see that $a^{-1}\in C^*(a)$ holds, since $\iota$ is invertible in $C(\sigma(a))$.