A class of archimedean lattice ordered algebras

Abstract

This paper introduces a 'new' class of lattice ordered algebras. A lattice ordered algebra A will be called a pseudo f -algebra if xy = 0 for all x, y in A such that x ∧ y is a nilpotent element in A. Different aspects of archimedean pseudo f -algebras are considered in detail. Mainly their integral representations on spaces of continuous functions, as well as their connection with almost f -algebras and f -algebras. Various characterizations of order bounded multiplicators on pseudo f -algebras are given, where by a multiplicator on a pseudo f -algebra A we mean an operator T on A such that xT (y) = yT (x) for all x, y in A. In this regard, it will be focused on the relationship between multiplicators and orthomorphisms on pseudo f -algebras.

Bidual of r-algebras

“…In [11], we studied a new class of lattice ordered algebras (so-called r-algebras) and presented its relation to the certain lattice ordered algebras [f -algebras [5] (a lattice ordered algebra A with the property that a^b D 0 implies that ac^b D ca^b D 0 for all c 2 A C /; almost-f -algebras [6] (a lattice ordered algebra A for which a^b D 0 in A implies that ab D 0/; d -algebras [9] (a lattice ordered algebra A such that a^b D 0 in A implies that ac^bc D ca^cb D 0 for all c 2 A C /; pseudo-f -algebras [7] (a lattice ordered algebra A having the property that ab D 0 if a^b is a nilpotent element of A/; and generalized almost-f -algebras [8] (a lattice ordered algebra A such that ab is an annihilator of A if a^b D 0/: A lattice ordered algebra A in which a^b D 0 in A implies that ab^ba D 0 is called an r-algebra. This is a wider class than both the classes of almost-f -algebras and d -algebras but, in general, independent of generalized almost-f -algebras.…”

On orthosymmetric bilinear maps

“…Contrary to L p ( p ≥ 3) , L 2 need not be a vector lattice under the ordering inherited from L as is proved in [7, Example 6.6]. If, however, L is a pseudo f -algebra then L 2 is a uniformly complete semiprime f -algebra (see [8,Proposition 4.4]). …”

“…For more information on order bounded multiplicators we refer the reader to [8]. An order bounded derivation on a semiprime f -algebra is necessarily identically zero.…”

“…Moreover, if L is, in addition, an almost f -algebra then so is L ru . In [8,Corollary 4.2.], Boulabiar and Hadded proved that with respect to this extended multiplication, L ru is a pseudo f -algebra whenever L is a pseudo f -algebra.…”

“…Now we briefly review some results on -algebras needed in the sequel. Our main references on the subject are [2,8,13]. A vector lattice L is said to be an -algebra if there exists an associative multiplication in L with the usual algebra properties such that f g ≥ 0 for all 0 ≤ f, g ∈ L. We denote by N (L) the set of all nilpotent elements of the -algebra L. We say that L is semiprime if N (L) = {0}.…”