Let $A$ be a complex, unital and commutative Banach-algebra.
Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
Background: By Gel'fand-Mazur, there is a continuous bijection from the spectrum $Sp(A)$ of $A$ (defined as the set of characters of $A$ endowed with the weak*-topology) onto $Max(A)$. The map sends a character to its kernel. I would like to know whether this bijection is a homeomorphism and this is equivalent to asking whether $Max(A)$ is Hausdorff. If $A$ is a $C^*$-algebra, then the answer is 'yes', because then $A$ is the ring of complex, continuous functions on $Sp(A)$ (by Gel'fand-Naimark) and one can use partition of unity to get the Hausdorff property of $Max(A)$