Irreducible faithful representation of $C^*$ algebras

Question

What is a terminology for those $C^*$ algebras which admit a faithful irreducible representation in some Hilbert space? Apart from simple algebras, what other algebras satisfy this property?

Answer 1

An ideal of a $C^*$-algebra $A$ is called "primitive" if it coincides with the kernel of some irreducible representation.

For commutative $C^*$-algebras the primitive ideals are precisely the maximal ideals and for this reason the study of primitive ideals is a bit like the study of Gelfand's transform. For example, people often consider the "primitive spectrum" of $A$, namely the set of all primitive ideals with a certain topology, and this space is often used to study $A$ more or less like the Gelfand spectrum (although with very limited success).

A $C^*$-algebra is called "primitive", as in Qiaochu Yuan's comment, if the zero ideal is primitive which is obviously the same as saying that said algebra admits a faithful irreducible representation.

Clearly every simple $C^*$-algebra is primitive, as noticed in the OP, but there are many primitive algebras which are not simple. For example the Toepliz algebra (the $C^*$-algebra generated by the one-sided shift) is primitive because its defining representation is ireducible (recall that it contains all compact operators) but it is not simple (again because it contains the ideal of compact operators).

An ideal $I$ of a $C^*$-algebra is called "prime" if given any two ideals $J$ and $K$ such that $J\cap K\subseteq I$, one has that either $J\subseteq I$ or $K\subseteq I$. It is not hard to show that every primitive ideal is prime (try it!) and the converse is true for separable $C^*$-algebras (a deep result).