If $A$ is a unital complex commutative Banach algebra to show that the Gelfand spectrum $\Omega (A)$ is weak star closed how to finish the following arguemnt:
My idea was to consider $\tau_n \in \Omega (A)$ such that $\tau_n \to \tau $ pointwise (=in weak star topology) and then prove that $\tau \in \Omega(A)$. The problem is that it seems obvious that $\tau$ is linear and multiplicative (and therefore in $\Omega (A)$) but the proof somehow needs to use that $A$ is unital otherwise it would be a redundant assumption. So I am probably missing something.
Later I came up with a proof which does use $1$.
But I am interested in finishing my first attempt. Given $\tau_n \to \tau$ as above is it possible to show $\tau \in \Omega (A)$?