State Spaces and the Kadison-Singer Property

Abstract

We introduce states and pure states for arbitrary C$^*$-algebras and discuss some of its fundamental properties. We generalize the concept of state extensions for matrix algebras as discussed in the previous chapter to the case of arbitrary unital abelian C$^*$-subalgebras of an operator algebra B(H), where H is some Hilbert space. Subsequently, we show that any pure state on the subalgebra that has a unique pure state extension, also has a unique state extension. In the case that every pure state on the subalgebra has a unique extension, we say that the subalgebra has the Kadison-Singer property.