Extensions, restrictions, and representations of states on 𝐶*-algebras

Abstract

In the first three sections the question of when a pure state g on a C ∗ {C^{\ast }} -subalgebra B of a C ∗ {C^{\ast }} -algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf ‖ b ( a − f ( a ) 1 ) b ‖ = 0 \left \| {b\left ( {a\, - \,f\left ( a \right )1} \right )b} \right \|\, = \,0 for each a in A, where the inf is taken over those b in B such that 0 ⩽ b ⩽ 1 0\, \leqslant \,b\, \leqslant \,1 and g ( b ) = 1 g(b) = 1 . The special cases where B is maximal abelian and/or A = B ( H ) A\, = \,B\left ( H \right ) are treated in more detail. In the remaining sections states of the form T ↦ lim u ( T x α , x α ) T \mapsto \lim \limits _{\mathcal {u}} \left ( {T{x_\alpha },\,{x_\alpha }} \right ) , where { x α } α ∈ κ \left \{ {{x_\alpha }} \right \}{\,_{\alpha \, \in \,\kappa }} is a set of unit vectors in H and u \mathcal {u} is an ultrafilter are studied.