Exercise 8, § 3

Let A be a commutative unital Banach algebra such that $\|1\|= 1$. Let N be the set of norms equivalent to the given norm, for which A is still a Banach algebra, and for which 1 has norm 1.

a) Let $x_0\in A$ such that $\varrho (x_0)<1$. For every $x\in A$, put:

$\|x\|'=$ inf $\sum_n\|a_n\|$

for all representations of $x$ in the form $a_0+a_1x_0+\cdots +a_nx^n_0$ (with $a_0, a_1, . . . , a_n\in A$). Now prove that $x\mapsto \|x\|'$ is an element of N. (Since $\varrho (x_0)<1$, there exists $k$ such that $\|x^n_0\|\leqslant k$ for all $n$, whence $\|x\|\leqslant k\|x\|'.$)

b) Now prove that $\|x_0\|'\leqslant 1$.

c) Deduce from a) and b) that, for every $x\in A$, one has

$\varrho (x) =$ inf$_{n\in N}n(x)$.

d) If the only element of N is the given norm, one has $A =\mathbf{C}\cdot 1$. (Use c) and exerc. 7.)