Pure state extensions and compressibility of the $l_ 1$-algebra

Abstract:

In 1958 Kadison and Singer proved that not every pure state on a continuous maximal abelian subalgebra (masa) of the algebra $\mathcal {B}(\mathcal {H})$ of all bounded linear operators on a separable complex Hilbert space $\mathcal {H}$, has a unique pure state extension to $\mathcal {B}(\mathcal {H})$ [5]. They conjectured the same result is true for discrete masas, and although the question remains open, it was shown by Anderson in 1978 to be equivalent to determining whether all operators in $\mathcal {B}(\mathcal {H})$ are compressible. We define in this paper the ${l_1}$-subalgebra $\mathcal {M}$ of $\mathcal {B}(\mathcal {H})$, and show that all operators in $\mathcal {M}$ are compressible. Hence every pure state on a discrete masa has a unique pure state extension to $\mathcal {M}$.