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Just-infinite   $C^*$  -algebras, that is, infinite dimensional   $C^*$  -algebras, whose proper quotients are finite dimensional, were investigated in [ 3 ]. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in [ 3 ]. In this article, we extend that construction by showing that each infinite dimensional metrizable Choquet simplex is affinely homeomorphic to the trace simplex of a just-infinite residually finite dimensional   $C^*$  -algebra. The trace simplex of any unital residually finite dimensional   $C^*$  -algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [ 3 ], and we show that it has precisely one extremal trace of type II  $_1$  .

Just-infinite $\boldsymbol{C^*}$ -algebras and Their Invariants