  By analogy with the well-established notions of just-infinite groups and
just-infinite (abstract) algebras, we initiate a systematic study of
just-infinite C*-algebras, i.e., infinite dimensional C*-algebras for which all
proper quotients are finite dimensional. We give a classification of such
C*-algebras in terms of their primitive ideal space that leads to a trichotomy.
We show that just-infinite, residually finite dimensional C*-algebras do exist
by giving an explicit example of (the Bratteli diagram of) an AF-algebra with
these properties.
  Further, we discuss when C*-algebras and *-algebras associated with a
discrete group are just-infinite. If $G$ is the Burnside-type group of
intermediate growth discovered by the first named author, which is known to be
just-infinite, then its group algebra $C[G]$ and its group C*-algebra $C^*(G)$
are not just-infinite. Furthermore, we show that the algebra $B = \pi(C[G])$
under the Koopman representation $\pi$ of $G$ associated with its canonical
action on a binary rooted tree is just-infinite. It remains an open problem
whether the residually finite dimensional C*-algebra $C^*_\pi(G)$ is
just-infinite.
