Closure properties of order continuous operators (Q1083675)

Let X be a compact Hausdorff space and let C(X) (or simply C) be the space of all real valued continuous functions of X. C'(X) and C''(X) (or C',C'' respectively) represent the first and second norm duals of C(X). C may be imbedded in C''. U is the set of limits in C'' of order convergent nets from C. If \(L^ r(E,F)\) and \(L^ c(E,F)\) represent respectively the spaces of regular and order continuous operators from Riesz space E to Riesz space F, then \(L^ r(C,C)\) and \(L^ r(C,U)\) may be imbedded in \(L^ c(C'',C'')\). We show that under this imbedding, both \(L^ r(C,C)\) and \(L^ r(C,U)\) are dense in the band which they generate in \(L^ c(C'',C'')\) in the topology defined (for fixed \(\mu \in C_+')\) by \[ \| T\|_{\mu}=<\mu,| T| 1>,\quad T\in L^ c(C'',C'') \] where 1 is the unit in C''. This is used to show that \(L^ r(C,C)\) and \(L^ r(C,U)\) are order dense in the band which they generate in \(L^ r(C'',C'')\). In addition, if X is a metric space, the band generated by \(L^ r(C,U)\) is all of \(L^ c(C'',C'')\).