We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space?
It seems enough to find all homomorphisms from the semigroup $S$ to unit disk $\mathbb D$, but what is the topology of maximal ideal looks like? Can we identify the topology of this maximal ideal with the product topology of $\prod \mathbb D$ ?