On some automorphisms of a class of Kadison–Singer algebras

Let L be the Kadison–Singer lattice generated by a nontrivial nest N on an infinite-dimensional separable Hilbert space H and a rank one projection Pξ determined by a separating vector ξ for the von Neumann algebra N″, and be the corresponding Kadison–Singer algebra. In this paper, we study the problem on the weak-∗ density of the subalgebra generated all the rank one operators, characterize the single elements in , and give the (quasi-)spatiality of an automorphism of , depending on whether I has an immediate predecessor in N or not.