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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 $\tau \in \Omega (A)$?

Student
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  • Hint: $\Omega(A)$ is defined as the set of nonzero multiplicative functionals. – Nate Eldredge Apr 11 '14 at 14:04
  • @NateEldredge Thank you , meanwhile I figured out a proof. Now I am interested in finishing my previous attempt. – Student Apr 11 '14 at 14:31
  • You are correct that $\tau$ will always be a multiplicative functional, but in the non-unital case it could be the zero functional. In the unital case you can rule that out. – Nate Eldredge Apr 11 '14 at 14:59

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