Exercise 22, § 2

Let A be a Banach algebra.

(a) Let S be the set of the bounded sequences of elements of A, with the norm $\|(x_n)\|=$ sup $\|x_n\|$. Let R be the set of the $(x_n)\in S$ such that $\|x_n\|$ tends to 0. Then S is a Banach algebra and R is a closed two-sided ideal of S.

Let $S'= S/R$ and $\varphi : S\rightarrow S'$ be the canonical morphism.

(b) The mapping $\theta$ of A into $S'$ defined by $\theta (x) =\varphi (x, x, x, . . .)$ is an isometric isomorphism of A onto a subalgebra of $S'$. Every left topological divisor of zero in $S'$ is a left divisor of zero. In order that $x\in A$ be a left topological divisor of zero, it is necessary and sufficient that $\theta (x)$ be a left divisor of zero in $S'$.

(c) Suppose that A has a unit element. Prove that there exist a Banach algebra B and an isometric isomorphism of A onto a subalgebra of B such that an element of A is non-invertible if, and only if, its image in B is a left or right divisor of zero. (Use a) and Exercises 19 and 21.)