The structure of Kadison-Singer lattices and the construction of strong Kadison-Singer algebras

Abstract

In this paper, we study the structure of Kadison-Singer lattices and strong Kadison-Singer algebras. We prove that every finite Kadison-Singer lattice on a finite-dimensional Hilbert space has no nontrivial reducing projections and construct a class of strong Kadison-Singer algebras on n-fold direct sums of infinite-dimensional separable Hilbert spaces. As a corollary, we show that there exist two Kadison-Singer algebras, such that their skew product is a strong Kadison-Singer algebra.