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.