Interpolation of states by vector states on certain operator algebras (Q1121495)

Let T be an operator on an infinite dimensional Hilbert space \({\mathcal H}\) with unit eigenvectors \(v_ i\) \((i=1,2,...)\). Suppose that \(\{v_ i\}\) forms a Schauder basis for \({\mathcal H}\) (i.e., for every \(x\in {\mathcal H}\), there is a unique sequence \(\{\alpha_ i(x)\}\) of scalars such that \(x=\lim_{n}\sum^{n}_{1}\alpha_ i(x)v_ i\) and such that \(x\to \alpha_ i(x)\) is continuous for every i). Let alg\(\{\) T,I\(\}\) be the weakly closed algebra generated by T and the identity I and let \(\phi\) be the linear functional on alg\(\{\) T,I\(\}\) given by \(\phi (S)=\sum c_ k<Sv_ k,v_ k>\) where \(c_ k>0\) and \(\sum c_ k=1\). Then the authors extend the results on finite matrices of \textit{D. Westwood} [J. Funct. Anal. 66, 96-104 (1986; Zbl 0587.47014)] by showing that there is a vector \(v\in {\mathcal H}\) such that \(\phi (S)=<Sv,v>\) for all \(S\in alg\{T,I\}\). Here two properties of Schauder bases \(\{x_ i\}\) of \({\mathcal H}\) (in general, reflexive spaces) are important: (1) given \(x\in {\mathcal H}\), then \(\lim_{n}\sup \{| <x,y>| | y\in sp\{x_ n,x_{n+1},...\}\}=0\) (the shrinking property); and (2) given \(\{\alpha_ n\}\) such that \(\sup_{n}| \sum^{n}_{1}\alpha_ ix_ i| <\infty\), then \(\{\) \(\sum^{n}_{1}\alpha_ ix_ i\}\) converges to an element in \({\mathcal H}\) (the boundedly complete property). The authors also present some examples and counterexamples.