  We consider the functor C that to a unital C*-algebra A assigns the partial
order set C(A) of its commutative C*-subalgebras ordered by inclusion. We
investigate how some C*-algebraic properties translate under the action of C to
order-theoretical properties. In particular, we show that A is finite
dimensional if and only C(A) satisfies certain chain conditions. We eventually
show that if A and B are C*-algebras such that A is finite dimensional and C(A)
and C(B) are order isomorphic, then A and B must be *-isomorphic.
