Let A be an AF algebra (i.e. a separable approximately finite-dimensional \(C^*\)-algebra.) Denote by M(A) the \(C^*\)-algebra of multipliers of A. In this note a description is given of the lattice of closed two sided ideals of M(A). More explicitly, it is established that
“The lattice of ideals of M(A) is isomorphic to the lattice of ideals of D(M(A))”.
Here D(M(A)) is the abelian local semigroup of equivalence classes of projections in M(A).
It is pointed out that determination of the ideal structure of D(M(A)) is still a problem; and the author discusses the case where A is simple.