LAUSR

Let A be an ordered algebra with a unit $$\mathbf{e}$$e and a cone $$A^+$$A+. The class of order continuous elements $$A_\mathrm{n}$$An of A is introduced and studied. If $$A=L(E)$$A=L(E), where E is a Dedekind complete Riesz space, this class coincides with the band $$L_\mathrm{n}(E)$$Ln(E) of all order continuous operators on E. Special subclasses of $$A_\mathrm {n}$$An are considered. Firstly, the order ideal $$A_\mathbf{e}$$Ae generated by $$\mathbf{e}$$e. It is shown that $$A_\mathbf{e}$$Ae 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 $$A^+$$A+ then $$A_\mathbf{e}$$Ae 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 $$C^*$$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 $$H^+$$H+. If the cone $$A^+$$A+ is normal then every element of $$A_\mathbf{e}$$Ae 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 $$T\in B(H)$$T∈B(H) is an order continuous element and operators $$T\in (B(H))_I$$T∈(B(H))I have properties which are analogous to the properties of orthomorphisms on Riesz spaces.