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Let A be a non-unital Banach Algebra.

$\Omega(A)=\{$set of all non zero multiplicative linear functionals of $A$ $\}$

I want to prove that $\Omega(A)\cup\{0\}$ is weak* compact. I've been given a hint that we have to show that the one point compactification of the gelfand spectrum is the set above and that would imply the result. I don't know how to go about proving it.

Moreover I was wondering how could I construct a sequence of elements in $\Omega(c_0(\mathbb R))$ such that it converges to $0$. Existence is true from the statement.

What I know is the fact that for a unital Banach algebra the Gelfand spectrum is weak* compact.

user141561
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    Just imitate the proof for unital Banach algebras; it's exactly the same. The hint you've been given is pretty useless as far as I can see. – Eric Wofsey Oct 07 '17 at 23:19
  • It is true that a sequence of multiplicative function would go to a multiplicative function. The proof I have read hugely relies on the fact that the Banach Algebra is unital to show that the limit functional is non zero. @Eric Wofsey – user141561 Oct 08 '17 at 09:06
  • Moreover can you hint me how to construct the sequence in $\Omega (c_0 (\mathbb R ))$ that I talked about? – user141561 Oct 08 '17 at 09:08

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