Exercise 31, § 6

Let A be the Banach space of integrable complex functions on $[0,1]$ for Lebesgue measure. We put

for $t\in [0,1]$.

a) The function $f\cdot g$ is defined almost everywhere and is an element of A.

b) Endowed with the operation $(f, g)\mapsto f\cdot g$, the space A is a commutative Banach algebra.

c) Let $x\in A$ be the constant function equal to 1. The function $x^n$ is

$(n-$ 1)!

d) The algebra A is topologically generated by $x$, the element $x$ is quasi-nilpotent, and A is equal to its radical.

e) The algebra A admits an approximate identity.

f) The algebra A possesses no maximal ideal, regular or not, closed or not (use the preceding exercise and Exercise 1 of I, p. 166). In particular, $\mathsf{X}(A)$ is empty, and hence compact.