I'm trying to get my head around Gelfand theory, and I can't seem to find the subtleties between commutative and non-commutative algebras. Why is there not a one-to-one correspondence between maximal ideals of a non-commutative algebra and the character homomorphisms from the algebra to the complex plane? Doesn't the Gelfand Mazur theorem apply to these algebras to, so if $\mathfrak{a}$ is maximal, the map
$$ A \to A/\mathfrak{a} \cong \mathbf{C} $$
is a homomorphism with kernel $\mathfrak{a}$. Conversely, if $\phi: A \to \mathbf{C}$ is a homomorphism, then $\phi$ is surjective, so if $\mathfrak{a} = \ker(\phi)$, $\tilde{\phi}: A/\mathfrak{a} \to \mathbf{C}$ is an isomorphism, hence $A/\mathfrak{a}$ is a field, so $\mathfrak{a}$ is maximal. What's going wrong here?
For a noncommutative algebra $A$ with maximal ideal $M$, $A/M$ need not be a field, so it is not always $\Bbb C$. It is just a simple ring, and those can get pretty unusual.