An embedding of the algebra of order bounded operators on a Dedekind complete Banach lattice (Q1193930)

Given a Dedekind complete Banach lattice, \(E\), we construct another Dedekind complete Banach lattice, \(\hat E,\) and an algebra homomorphism, \(T\mapsto \hat T\), of the order bounded linear operators on \(E\) into the bounded linear operators on \(\hat E\) such that the regular norm of \(T\) is equal to the operator norm of \(\hat T\) and the order spectrum of \(T\) is equal to the spectrum of \(\hat T.\) We deduce that the order spectrum of lattice homomorphisms and of positive interval preserving linear operators are cyclic.