Exercise 3, § 3

Let Λ be a set, $p\in [1,+\infty [$, and $A =\ell^p(\Lambda )$ the set of families $x= (x_{\lambda})_{\lambda\in\Lambda}$, where $x_{\lambda}\in \mathbf{C}$ and

a) For componentwise addition and multiplication, and the norm $x\mapsto \|x\|_p$, the space A is a Banach algebra.

b) One has $A^2\not= A$ and $A^2$ is dense in A.

c) The space $\mathsf{X}(A)$ of the nonzero characters of A is naturally identified with Λ endowed with the discrete topology. (Use the fact that the dual of $\ell^p(\Lambda )$ is $\ell^q(\Lambda )$, where $q$ is the conjugate exponent of $p.$) The Gelfand transform then becomes the identity mapping and its image in the space of continuous functions tending to 0 at infinity on Λ is dense and not closed.