If $A$ is a commutative $C^*$ $*$-algebra, then the kernel of a nontrivial character is a maximal ideal.
If $A$ is an arbitrary $C^*$-algebra, what kind of subspace is the kernel of a pure state?
The motivation is the following: in the commutative case, we have the $C^*$-algebra version of the Stone-Weierstrass theorem, in which the concept of "character" (or, equivalently: maximal ideal) is central; if the algebra is no longer commutative, the space of characters (the spectrum of the algebra) is replaced by the pure state space. I would therefore like to understand what the correspondence between characters and maximal ideals changes into when the underlying algebra is no longer commutative.