On some automorphisms of a class of Kadison–Singer algebras☆

Abstract 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 P ξ determined by a separating vector ξ for the von Neumann algebra N ″ , and alg L 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 alg L , and give the (quasi-)spatiality of an automorphism of alg L , depending on whether I has an immediate predecessor in N or not.