Interpolation of states by vector states on certain operator algebras

LetT be an operator on an infinite dimensional Hilbert space \(\mathcal{H}\) with eigenvectorsv _i , ‖v _i ‖=1,i=1, 2, ..., andsp{v _i ?i∈n} dense in \(\mathcal{H}\) . Suppose that {v _i } is a Schauder basis for \(\mathcal{H}\) . We denote byA _T the ultraweakly closed algebra generated byT andI, the identity operator on \(\mathcal{H}\) . For any nonnegative sequence of scalars \(\left\{ {\alpha ,with = \sum\nolimits_1^\infty {\alpha _1 } = 1} \right\},\) , we associate an ultraweakly (normal) continuous linear functional \(\phi _\alpha = \sum\nolimits_1^\infty {\alpha _j } \omega _v\) where \(\phi _\alpha \left( A \right) = \lim _n \sum\nolimits_1^n {\alpha _j } \omega _v ,\) , and \(\omega _v ,\left( A \right) =< Av_1 ,v_1 >\) for allA∈A _T . We denote the set of all such linear functionals onA TbyF(T). The question that we investigate in this paper is whether each linear functional φ_α inF(T) is a vector state, i.e. does φ_α=ωx for some unit vectorx in \(\mathcal{H}\) ?