On generators of abelian Kadison–Singer algebras in matrix algebras

Assume that H is a Hilbert space of dimension greater than two. We prove that an abelian Kadison–Singer algebra acting on H cannot contain any non-trivial idempotent. Based on this, we show that an abelian KS-algebra in matrix algebra M n ( C ) ( n ⩾ 3 ) cannot be generated by a single element. As a corollary, it is also proved that the lattice of an abelian KS-algebra cannot be completely distributive.