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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?

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Using nets does not generate any problem. Suppose that $f_j\to f$ pointwise in $M(A)$. Given $x,y\in A$, $$ f(xy)=\lim f_j(xy)=\lim f_j(x) \,f_j(y)=f(x)f(y). $$ The non-obvious equality is the last one. The only difference with the case of sequences is that a convergent net need not be bounded; but it is eventually bounded, and so the proof that the limit of the product is the product of the limit is the same as in the case of sequences.

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