An almost f-algebra multiplication extends from a majorizing sublattice

A lattice ordered algebra (E, • ) is called an almost f -algebra if x Л y = 0 implies x • y = 0 for all x,y G E or equivalently | x | • | x | = x • x for every x G E (cp. [2]). C. B. Huijsmans in [6] posed the question of whether the multiplication of an almost f -algebra can be extended to its Dedekind completion. G. Buskes and A. van Rooij in [4, Theorem 10] answered in the affirmative to the question and this result raises naturally another question: Can an almost f-algebra multiplication given on a majorizing vector sublattice of a Dedekind complete vector lattice be extended to an almost f -algebra multiplication on the ambient vector lattice? A positive answer was announced in [8, Corollary 7] by the author:

Theorem. Let E be a ma jorizing sublattice of a Dedekind complete vector lattice E ^b and simultaneously an almost f -algebra. Then E can be endowed with an almost f algebra multiplication that extends the multiplication on E .

The aim of this note is to present the proof. Our reasoning is along the same lines as in [4] and rely upon a general structure theorem for almost f -algebras saying that they are actually distorted f -algebras as was shown in [4, Theorem 2]. All vector lattices and lattice ordered algebras are assumed to be Archimedean.

Recall that an f -algebra is a lattice-ordered algebra whenever x Л y = 0 implies (a • x) Л y = 0 and (x • a) Л y = 0 (or equivalently (x • y) Л a = 0 provided that x Л a = 0 or y Л a = 0) for all x,y G A and a G A+. It is well known that an f -algebra multiplication is commutative [2] and order continuous [9].

For an arbitrary vector lattice E there exists a (essentially unique) pair (E ⁰ , 0 ) such that E ⁰ is a vector lattice, 0 is a symmetric lattice bimorphism from E x E to E ⁰ and the following universal property holds: for every symmetric lattice bimorphism b from E x E to some vector lattice F there exists a unique lattice homomorphism Ф ь : E ⁰ ^ F with b = Ф ь 0 . This notion was introduced by G. Buskes and A. van Rooij, see [5] and [3]. The said universal property remains valid if we replace b and Ф ь by a positive orthosymmetric ( = x Л y = 0 ^ b(x, y) = 0) bilinear operator and a positive linear operator provided that F is uniformly complete, see [5, Theorem 9] and [3, Theorem 3.1].

An almost f -algebra multiplication extends from a ma jorizing sublattice

We now present the needed structure result from [4, Theorem 2]. Let E be an arbitrary vector lattice and h a lattice homomorphism from E into a Dedekind complete semiprime f -algebra G with a multiplication o . Then F := h(E) is a sublattice of G and in view of [3, Proposition 2.5] F ⁰ can be considered also as a sublattice of G with (x,y) ^ x o y instead of 0 . Denote by F ⁽²⁾ the linear hull of { x o y : u,v £ F } . Then F ⁰ is the sublattice of G generated by F ⁽²⁾ and F ⁽²⁾ is uniformly closed in F ⁰ .

Let a positive linear operator Ф from F ⁽²⁾ to E and an element ш £ G are such that hФ(u) = ш o u for all u £ F ⁽²⁾ . Of course, one can consider Ф as a positive operator from F ⁰ to E ^ru , the uniform completion of E. Put x • y : = Ф(hx o hy) (x, y £ E). Then (E, • ) is an almost f -algebra. Indeed, (x, y) ^ x • y can be taken as an almost f -algebra multiplication, since evidently x Л y = 0 implies hx o hy = 0, whence x • y = 0 and its associativity is also easily seen:

(x • y) • z = Ф(ш o hx o hy o hz) = x • (y • z).

It is proved in [4, Theorem 2] that every Archimedean almost f -algebra arises in this way.

C Proof of the Theorem. Let E be a majorizing sublattice of a Dedekind complete vector lattice E and also an almost f -algebra under a multiplication • . According to [4, Theorem 2] one can choose F , G, o , h, and Ф as above. By [1, Theorem 7.17] or [7, Theorem 3.3.11 (2)] there exists a lattice homomorphism h from E onto a sublattice F C G extending h. Moreover, F is a majorizing and order dense sublattice of F . By [3, Proposition 2.7] F ⁰ is also a majorizing and order dense sublattice of (F ) ⁰ . According to [1, Theorem 2.8] or [7, Theorem 3.1.7] the positive operator Ф from F ⁰ to E ^ru C E has a positive extension Ф from(F) ⁰ to E . Now, h Ф is obviously an extension of h Ф and it remains to ensure that hФ(u) = ш o u (u £ (F) ⁰ ), since in this event an almost f -algebra multiplication on E can be defined by x • y : = Ф(h(x) o h(y)) (x, y £ E) as was observed above. For a fixed u £ (F) ⁰ take arbitrarily u ⁰ , u ⁰ £ F ⁰ such that u ⁰ 6 u 6 u ⁰⁰ . Then ш o u' = hФ(u ⁰ ) 6 h Ф(u) 6 h Ф(u ⁰⁰ ) = ш o u ⁰⁰ and thus, by order continuity of f -algebra multiplication, 8ир { ш o u ⁰ } = h Ф(u) = inf { ш o u ⁰⁰ } = ш o u. B