On orthomorphism elements in ordered algebra

Let C be an ordered algebra with a unit e . The class of orthomorphism elements Orthe(C) of C was introduced and studied by Alekhno in ”The order continuity in ordered algebras”. If C = L(G) , where G is a Dedekind complete Riesz space, this class coincides with the band Orth(G) of all orthomorphism operators on G . In this study, the properties of orthomorphism elements similar to properties of orthomorphism operators are obtained. Firstly, it is shown that if C is an ordered algebra such that Cr , the set of all regular elements of C , is a Riesz space with the principal projection property and Orthe(C) is topologically full with respect to Ie , then Be = Orthe(C) holds, where Be is the band generated by e in Cr . Then, under the same hypotheses, it is obtained that Orthe(C) is an f -algebra with a unit e .


Introduction
All vector spaces are considered over the reals only. An ordered vector space (Riesz space) C under an associative multiplication is said to be an ordered algebra (Riesz algebra) whenever the multiplication makes C an algebra, and in addition it satisfies the following property: a, b ∈ C + implies ab ∈ C + . A Riesz algebra C is called an f -algebra if C has the additional property that a ∧ b = 0 implies ac ∧ b = ca ∧b = 0 for each c ∈ C + .
Throughout the study, we will assume C ̸ = {0} and C has a unit element e > 0 . An element a ∈ C is called a regular element if a = b − c with b and c positive, the space of all regular elements of C will be denoted by C r . Obviously, C r is a real ordered algebra. Let C be an ordered vector space and an element a ∈ C + , the order ideal I a generated by a is the set I a = {b ∈ C : −λa ≤ b ≤ λa for some λ ∈ R + } . Under the algebraic operations and the ordering induced by C , I a is an ordered vector subspace of C . Moreover, I e is an ordered algebra [1].
An element q ∈ C is said to be an order idempotent whenever 0 ≤ q ≤ e and q 2 = q . Under the partial ordering induced by C , the set of all order idempotents OI(C) of C is a Boolean algebra and its lattice operations satisfy the identities p ∧ q = pq and p ∨ q = p + q − pq for all p, q ∈ OI(C). If c ∈ C and the modulus |c| of c exists, then q |c| = |qc| and |c| q = |cq| for all q ∈ OI(C) [2] . Definition 1.1 [1] Let C be an ordered algebra, an element a ∈ C is said to be an order idempotent preserving element whenever (e − q)aq = 0 for all q ∈ OI(C) . An element a is said to be an orthomorphism element of an ordered algebra C whenever a is an order idempotent preserving element that is also regular.
The collection of all orthomorphism elements of an ordered algebra C will be denoted by Orthe(C). An operator π : G → G on a Riesz space G is said to be band preserving whenever π(B) ⊆ B holds for each band B of G . π is a band preserving operator if and only if π(x) ⊥ y whenever x ⊥ y in G . A band preserving and order bounded operator π is called orthomorphism of G and the set of all orthomorphisms of G is denoted by Orth(G) . If G has the principal projection property, then an operator π : G → G is band preserving if and only if πp = pπ (or (I − p)πp = 0 ) for every order projection p on G [3,Theorem 8.3] . If C = L(G) is taken, where G is a Dedekind complete Riesz space, then the set of all order idempotents OI(C) of C is the set of all order projections on G [3, Theorem 3.10] and the band B e generated by e in C r is equal to Orth(G) = Orthe(C) [3,Theorem 8.11]. In general, the equality B e = Orthe(C) does not hold in the case of an arbitrary ordered algebra C . Therefore, the following question might come into mind. Under what condition Orthe(C) could be identified to B e ? In this work, we try to provide an answer to this question. Moreover, we will show that, under the same hypothesis, Orthe(C) has the similar properties of orthomorphisms.
We refer to [3,5,7,9] for definitions and notations which are not explained here. All Riesz spaces in this paper are assumed to be Archimedean.

Ortomorphism elements
where B e is the band generated by e in C r .
Proof Since q |a| = |qa| and |a| q = |aq| for all q ∈ OI(C) and a ∈ C r , it is easy to show that Orthe(C) is Lemma 2.2 Let C be an ordered algebra such that C r is a Riesz space with the principal projection property Freudenthal's Spectral Theorem [3, Theorem 6.8] , there exists a sequence (u n ) of e -step function satisfying 0 ≤ a − u n ≤ n −1 e for each n and u n ↑ a for every a ∈ I e . As u n e -step function, there exist λ 1 , λ 2 , ..., λ k ∈ R and p 1 , p 2 , ..., p k ∈ OI(C) such that for each n . Since C is Archimedean, we have ab = ba for every a ∈ I e . 2 is the generated by the identity operator I in C r . In general, the equality B e = Orthe(C) does not hold in the case of an ordered algebra C .  Let C be a Riesz algebra such that C r is a Riesz space. It is easy to see that (bc)q = q(bc) for each b, c ∈ Orthe(C) and q ∈ OI(C). Thus, Orthe(C) is a Riesz algebra. For b ∈ Orthe(C), let us define Proof Let b, c ∈ Orthe(C) and b ∧ c = 0. From the Lemma 2.6 we have Proof Let b ∈ Orthe(C) with |b| ∧ e = 0 . Clearly, holds for each h ∈ Orthe(C) ∼ + . Then, it follows that Proof It is clear that the band generated by e in Orthe(C) is equal to the band generated by e in C r as By the Example 2.5, we have known that if G is a Dedekind complete Riesz space with separating order dual and C = L(G), then Orthe(C) has separating order dual and Orthe(C) = Orth(G) is topologically full with respect to I e = Z(G). By using this observation and the above result, we can obtain the following Corollary being previously proved as a theorem in a different manner.

Corollary 2.10
Let G be a Dedekind complete Riesz space and G has separating order dual. Then the band B I generated by the identity operator in L r (G) is equal to Orth(G).

Theorem 2.11
If C is an ordered algebra such that C r is a Riesz space with the principal projection property and Orthe(C) is topologically full with respect to I e , then Orthe(C) is an f -algebra. Moreover, it is a full subalgebra of C .
Let b ∈ Orthe(C) be invertible in C . We will show that b −1 ∈ Orthe(C) . As b ∈ Orthe(C) bq = qb for each q ∈ OI(C) . It is easy to see that b −1 q = qb −1 for each q ∈ OI(C) . Thus, Orthe(C) is a full subalgebra of C . is an f -algebra. Moreover, it is a full subalgebra of L r (G).
As each unital f -algebra C with separating order dual is topologically full with respect to I e [8] , we can give the following corollary.

Corollary 2.13
Let C be an ordered algebra such that C r is a Riesz space with the principal projection property and Orthe(C) has separating order dual. Then, Orthe(C) is an f -algebra if and only if Orthe(C) is topologically full with respect to I e .
As we said before, if G is a Dedekind complete Riesz space with separating order dual and C = L(G) then Orthe(C) = Orth(G) is topologically full with respect to I e = Z(G) . However, even if C is a Dedekind complete ordered algebra, Orthe(C) may not be topologically full with respect to I e . We now give an example of a Dedekind complete ordered algebra which is not topologically full with respect to I e .
Example 2.14 Let f be a multiplicative functional on l ∞ satisfying f (c 0 ) = 0 and C be the linear space is not an f -algebra. By the Corollary 2.13, Orthe(C) is not topologically full with respect to I e .
Since each f -algebra is commutative, we can give the following corollary.

Corollary 2.15
Let C be an ordered algebra such that C r is a Riesz space with the principal projection property and Orthe(C) is topologically full with respect to I e . Then, Orthe(C) is a commutative algebra.