Let $\mathcal A$ be commutative unital Banach algebra denote by $M(A)$ the space of all non-zero multiplicative functionals on $\mathcal A$.
I want to show that $M(A)$ is closed in the weak-* topology.
Now, for a general, non-separable $\mathcal A$, one has to use nets, i.e. $M(\mathcal A)$ is closed if and only if it contains the limit points of all its nets.
How can one technically deal with those nets, i.e. how can one show that the multiplicative property is still valid in the limit?