The order continuity in ordered algebras

Let A be an ordered algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). The class of order continuous elements \(A_\mathrm{n}\) of A is introduced and studied. If \(A=L(E)\), where E is a Dedekind complete Riesz space, this class coincides with the band \(L_\mathrm{n}(E)\) of all order continuous operators on E. Special subclasses of \(A_\mathrm {n}\) are considered. Firstly, the order ideal \(A_\mathbf{e}\) generated by \(\mathbf{e}\). It is shown that \(A_\mathbf{e}\) can be embedded into the algebra of continuous functions and, in particular, is a commutative subalgebra of A. If A is an ordered Banach algebra with normal cone \(A^+\) then \(A_\mathbf{e}\) is an AM-space and is closed in A. Secondly, the notion of an orthomorphism in the ordered algebra A is introduced. Among others, the conditions under which orthomorphisms are order continuous, are considered. In the second part, the main emphasis will be on the case of an ordered \(C^*\)-algebra A and, in particular, on the case of the algebra B(H), where H is an ordered Hilbert space with self-adjoint cone \(H^+\). If the cone \(A^+\) is normal then every element of \(A_\mathbf{e}\) is hermitian. In H the operations are introduced which coincide with the lattice ones when H is a Riesz space. It is shown that every regular \(T\in B(H)\) is an order continuous element and operators \(T\in (B(H))_I\) have properties which are analogous to the properties of orthomorphisms on Riesz spaces.