Exercise 3, § 4

Let $A$ be a unital Banach algebra, $G$ the group of invertible elements of $A$ and $G_1$ the identity component of G.

a) Let $x\in A$ and $n$ an integer $\geqslant 0$ such that $x^n=e$. Now prove that $x\in G_1$. (The spectrum of $x$ is finite. The set of the $\lambda \in \mathbf{C}$ such that $y(\lambda ) =\lambda x+e-\lambda e$ is invertible has finite complement, hence is connected. Now, $y(0) =e,y(1) =x.$) b) Suppose that A is commutative. Every element of $G/G_1$ distinct from the identity element is of infinite order. (If $x\in G$ and $x^n\in G_1$, one has $x^n=$ exp($y$) with a $y\in A$ according to No.$^o10$ of I, p. 78, whence $(x$ exp($-y/n$))$^n=e$. Apply a).)