Regular States and the Regular Algebra Numerical Range

Abstract

Let E be a Dedekind complete complex Banach lattice and let $\mathcal L_{r}(E)$ denote the Banach lattice algebra of regular operators on E. Then a bounded linear functional $\Phi :\mathcal L_{r}(E)\to \mathbb C$ is called a regular state if Φ(I) = ∥ Φ∥ = 1. Basic properties of regular states are derived and a detailed description of order continuous regular states is obtained for ℓ _p-spaces. The regular states define the regular numerical algebra range $V(\mathcal L_{r}(E), T)=\{\Phi (T): \Phi \mbox{ a regular state}\}$. We describe the regular algebra numerical range for operators T in the center Z(E) of E and for operators T disjoint with the center. Using the description of regular states on ℓ _p(n), we characterize the regular numerical range preserving linear operators on $\mathcal L_{r}(\ell _{p}(n))$.

Dedicated to Ben de Pagter on the occasion of his retirement