Exercise 13, § 3

Let A be the Banach algebra $\ell^2(\mathbf{N})$ (Exercise 3). Let $A_0$ be the subalgebra formed by the sequences of finite support. Let B be the direct sum of $A_0$ and $\mathbf{C}$, equipped with the multiplication $(f, \alpha )(g, \beta ) = (f g,0)$ for $f, g\in A_0,\alpha , \beta \in \mathbf{C}$, and with the norm

$\|(f, \alpha )\|=$ sup$(\|f\|,|\alpha -\sum_nf(n)|)$.

a) The completed algebra $\widehat{B}$ is a commutative Banach algebra whose radical R is $\mathbf{C}\cdot (0,1)$, and $\widehat{B}/R$ is isometrically isomorphic to A.

b) Let $\pi :\widehat{B}\rightarrow \widehat{B}/R$ be the canonical morphism. There exists no subalgebra $B_1$ of $\widehat{B}$ supplementary to R such that $\pi |B_1: B_1\rightarrow \widehat{B}/R = A$ is a homeomorphism. (Let $B_1$ be such a subalgebra. Let $u_n\in A$ such that $u_n(n) = 1,u_n(m) = 0$ for $m\not=n$. Let $e_n\in B_1$ such that $\pi (e_n) =u_n$. Now prove

$\infty \infty$

1 1

that $e_n\in A_0$, that $\sum u_n$ converges in A but that $\sum e_n$ does not converge

$n=1n=1$

in $\widehat{B}.)$