Kadison Transitivity (Used in §6)

A form of Kadison’s theorem used in §6: if $\mathfrak A$ acts irreducibly on $\mathcal H$ and $x,y\in\mathcal H$, then there exists $A\in\mathfrak A$ with $Ax=y$.

Key property (paper use):

• Lets Shale characterize “relative normalizability” of states via $F(X)=E(A^*XA)$.

Example: In $B(\mathcal H)$, take $A$ to be a rank-one operator sending $x$ to $y$.