Cone domains separate FS-domains from RB-domains

Abstract

Let $C$ be a closed, convex, pointed and generating cone in a finite-dimensional real vector space $V$, and let \( D_C=(-C)\cup\{\bot\}\) be the negative cone with a new least element, ordered by the cone order. Keimel proved that these cone domains are FS-domains and asked whether they are always retracts of bifinite domains. We give a sharp answer: \[D_C\text{ is an RB-domain}\quad\Longleftrightarrow\quad C\text{ is simplicial}. \] Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. The proof converts the RB approximation property into finite-valued $C$-isotone approximations of the identity. The analytic obstruction is elementary and finite-dimensional: first in Euclidean space, cone-upper sets are represented, up to null sets, as Lipschitz epigraphs; Rademacher's theorem, Fubini's theorem and integration by parts then force the matrix tested against any finite-valued isotone map to lie in the cone generated by the positive rank-one operators $v\otimes \ell$, $v\in C$, $\ell \in C^{\ast }$. If such maps approximate the identity, the identity operator lies in this rank-one cone, which is possible exactly when the cone is simplicial. This answers Keimel's question in the negative for the Lorentz cone and other non-simplicial cones.