Pure State Extensions in Linear Algebra

Abstract

We introduce the concept of the Kadison-Singer property by means of a very concrete and accessible example. We define the concept of a state for both a matrix algebra and its subalgebra consisting of diagonal matrices. Subsequently, we define pure states and give explicit descriptions of states and pure states in these concrete examples. We finish by showing that any pure state on the algebra of diagonal matrices can be uniquely extended to a pure state on the whole matrix algebra. In the chapters that follow, it will be made clear that this precisely means that the algebra of diagonal matrices has the Kadison-Singer property.