NEW ZEALAND JOURNAL OF MATHEMATICS Volume 30 (2001), 25-29

A N O TE ON TH E C O M M U T A T IV IT Y OF A R C H IM E D E A N A L M O ST /-R I N G S

G e r a r d B u s k e s a n d J a s o n H o l l a n d

(Received February 1999)

Abstract. We prove, via orthosymmetric maps, that every Archimedean al­most /-r in g is commutative.

In the theory of partially ordered structures, vector lattices (also called Riesz spaces) and lattice ordered groups have been studied extensively (see e.g. [1]). Indeed, examples of vector lattices appear at many places in classical functional analysis and the study of lattice ordered groups goes back to the beginning of this century. Less attention has been paid to lattice ordered algebras and lattice ordered rings, and the history of that attention is more recent. Nonetheless, for the study of positive operators and the structure theory of vector lattices, Archimedean /-rings have played an important role in the last decades. Their properties, often due to the order continuity of the multiplication, are well understood. They are commutative, their multiplication uniquely extends to an /-ring multiplication on the Dedekind completion and they appear in a variety of theories, from spectral measures to the von Neumann bicommutant theorem.

Almost /-rings, i.e. lattice ordered rings in which

/ A g = 0 implies that fg = 0,

are not as well behaved and less commonly found in applications. However, Archimedean almost /-algebras also are commutative (see [2], [3] and [6]) and their multiplication can be extended, though not uniquely, to the Dedekind completion (see [4]). Though the latter property does not carry over to Archimedean almost /-rings as we will show next, more surprisingly the former does. For unexplained terminology we refer to [1].

Example 1. In this example, c is the space of all real-valued convergent sequences and for each f £ ewe denote the limit of the sequence / by / ( oo). Furthermore, 1 is the sequence that is everywhere equal to 1 and 0 is the sequence that is everywhere equal to 0.

Let G = { / £ c : / (0 ) , /(o o ) £ Z }. It follows that the Dedekind completion of G, denoted by G6, equals { g e l 0 0 : g( 0) £ Z }. We equip G with the following multiplication:

f - 9 ■= f{oo)g(oc ) l (f,g<EG).

2000 A M S Mathematics Subject Classification: 06F25.

26 GERARD BUSKES AND JASON HOLLAND

Suppose that this multiplication were to extend to Gs , again denoted by a period. Let

x = ’ y = ’ and ̂=

Then 1 = 1 - (x + y) = l - x + l - y. Since x — z + y, we have1 = 1 • (z + y) + 1 -y

= 0 + 1 -y + l - y = 1 -y + l - y.

However, (1 -y)(0) = \ contradicts the fact that (1 -y)(0) G Z, since 1 - y is assumed to be an element of G6.

The commutativity of almost /-algebras has been well-known for a while (see [2] and [6]). The proof in [3] of that commutativity employed the automatic symmetry of orthosymmetric maps rather than the multiplication. That road provides an easy avenue for the result in the title of this paper. We first need the obvious generalization of orthosymmetric maps to lattice ordered groups. If F and G are groups, A : G x G —* F is called a group bimorphism if yi is a group homomorphism in each coordinate. For Archimedean /-groups F and G, a map A : G x G —► F is called orthosymmetric, if the following hold:(i) if / , g G G and / A g = 0, then A(f ,g) = 0, and

(ii) A is a group bimorphism.The map A is called positive, if for every / , g G G+ , it follows that A(f, g) G F + .

When applied to the special situation that F and G are Archimedean Riesz spaces, the definition of orthosymmetric map reduces to the definition given in [3]. We state Corollary 2 from [3] to which we will reduce the main result of this paper.Theorem 2. Let E and F be Archimedean vector lattices. Let A be an orthosym­metric positive bilinear map E x E —> F. Then A ( f , g ) = A(g , f ) for all f , g G E.

Let G be a divisible lattice ordered subgroup of a lattice ordered group H. Let e G H, u G G. Suppose that (gn)nen is a sequence of elements of G. We will consider G as a vector space over the rationals. We say that gn converges to e (notation gn —» e) with respect to u if there exists a sequence (£n)neN ° f rational numbers decreasing to zero and u g G , such that \gn — e\ < £nu for all n G N. If for every e G H there exists a sequence {gn)neN °f elements of G and u G G for which gn —► e with respect to u, then we say that G is relatively uniformly dense in H. For an Archimedean /-group G, we denote the divisible hull of G by Gd. Clearly, Gd is relatively uniformly dense in the vector lattice cover of G (see [5]), denoted by R[G]. On its turn, i?[G] is relatively uniformly dense and majorizing in its relative uniform completion, which we denote by R[G]ru.

Of course, the multiplication in an almost /-ring is the main example of a positive orthosymmetric map. Our approach then, for a given Archimedean almost /-ringG, will extend any orthosymmetric map on G , first to Gd and then to R[G]ru. The first of these steps is straightforward and we leave its proof to the reader.Lemma 3. Let F and G be Archimedean I-groups and let A : G x G —► F be a positive orthosymmetric map. Then A extends uniquely to a positive, orthosym­metric map A : Gd x Gd —> F d.

A NOTE ON THE COMMUTATIVITY OF ARCHIMEDEAN ALMOST /-R IN G S 27

In fact if A in the previous Lemma is derived from an almost /-algebra multi­plication then so is A. To extend to R[G]ru we need the following lemma.

Lemma 4. Let G be a divisible, Archimedean l-group that is relatively uniformly dense in the lattice ordered group E. Then for every e G E+ there exists a sequence (/n)nen in G for which f n —̂ e and for which f n < e for every n G N.

Proof. Let e G E +. Let (gn)nen be a sequence in G, such that gn —> e with respect to u G G. There exists a sequence (en)neN of rational numbers decreasing to zero, such that |gn — e\ < £nu for all n G N. Now define

fn •— Ifl'n) £nNote that for every n G N,

I \gn\ - e| < 19n - e| < £nu,

which implies that f n = \gn\ — £nu < e for every n G N. Then

| fn = | \9n\ £ n ^ c| ^ | |<?n| “h £ nU fC £ n ^ “t” 2£ n U

for every n G N. Thus f n —> e. □

Theorem 5. Let G be an Archimedean, divisible Ingroup such that G is relatively uniformly dense and majorizing in an Archimedean Riesz space E. Let F be a uni­formly complete Riesz space and let A : G x G —> F be positive and ortho symmetric. Then A extends uniquely to a positive, bilinear ortho symmetric map A : E x E —* F.

Proof. Let f ,g G E. By the relative uniform density of G in E, there exist sequences ( /n)nen and {9n)nen of elements of G, u\, U2 G E , and a sequence (£n)neN of rational numbers decreasing to zero, such that | f n — f | < enui, and |gn — g\ < enU2■ It follows that \fn\ < £\U\ + |/| and |grn| < £1 2̂ + Ifi1! for every n G N. Since G is majorizing in E , there exists h G G such that

h > ui V u2 V (eiui + I/I) V (eiu2 + |p|).Then for n > m,

\A{fn,9n) - A(fm,gm)\ < \ M\fn ~ fm |, \gn I) | + |̂ ( I/m |, \9n ~ 9m |) |< A {2£nh,h) + A{h,2enh) = 4;£nA{h,h).

Thus (A(fn, 9n))nen is a relative uniform Cauchy sequence in F and has a limit in F. If also f n —> / and 9 ̂ —> g in the same sense as our f n —> f and gn —> g above, then (A(fn,g^) - A ( fn,gn))neN converges relatively uniformly to zero by a computation similar to the above. Define

A(f ,9) ■= lim A ( fn,gn) ( f ,g G E).n—>00

It is straightforward to prove that A is positive and bilinear. It remains to show, that A is orthosymmetric. To that end, let f , g E E such that / A g = 0. By Lemma 4, there exist sequences ( /n)nen and (gn)nen °f elements of G such that f n —> / , gn —̂ 9, fn < / for all n G N and gn < g for all n G N. Then A(f ,g) = limn_+00 A ( fn,gn) = 0. □

The following Corollary is now an immediate consequence of the automatic sym­metry of orthosymmetric maps on vector lattices (see [3]).

28 GERARD BUSKES AND JASON HOLLAND

Theorem 6 . Every Archimedean almost f-ring is commutative.

P roof. Let G be an Archimedean almost /-ring. First extend the multiplication to an almost /-ring multiplication on the divisible hull Gd. The latter multiplication can be seen as a positive orthosymmetric map A : Gd x Gd —> Gd. Now embed Gd in the vector lattice cover R[G] and embed R[G] in its uniform completion R[G]ru. Interpret the map A as a map with values in R[G]ru. The map A can be extended to an orthosymmetric positive and linear map

A : R[G]ru x R[G]ru -* R[G]ru,

by applying the previous theorem twice. Now apply Theorem 2. □

Remarks.1. Natural as the approach above may seem, it is not clear that the extension

procedure above puts an almost /-algebra multiplication on R[G]ru. Indeed, it is the associativity of the only candidate that is a problem. If G has an order unit then Gd is relatively uniformly dense in R[G]ru and the procedure outlined above does extend the multiplication on Gd in one step (avoiding R[G]) to an associative almost /-algebra multiplication on R[G]ru. Whether or not this order unit is needed is an open question. However, many algebraic properties known for Archimedean almost /-algebras can be carried over almost routinely to Archimedean almost /-rings, via the approach provided above.

2. We remark that the authors of [2] (on page 287) claim that their proof of the commutativity of almost /-algebras carries over to the setting of almost /-rings. Thus the main result (Theorem 6) of the present paper may not be new. However, the proof of the commutativity of almost /-algebras in [2] is rather involved and the reduction argument above offers a different way to verify the commutativity of almost /-rings while highlighting the elegance of using orthosymmetric maps.

References

1. M. Anderson and R. Feil, Lattice Ordered Groups, an Introduction, D. Reidel Publishing Co., 1987.

2. S.J. Bernau and C.B. Huijsmans, Almost f-algebras and d-algebras, Math. Proc. Cambridge Phil. Soc. 107 (1990), 287-308.

3. G. Buskes and A. van Rooij, Almost f-algebras: commutativity and the Cauchy- Schwarz inequality, Positivity and its Applications (Ankara, 1998), Positivity 4 No. 3 (2000), 227-231.

4. G. Buskes and A. van Rooij, Almost f-algebras: structure and the Dedekind completion, Positivity and its Application (Ankara, 1998), Positivity 4 No. 3 (2000V 233-243.

A NOTE ON THE COMMUTATIVITY OF ARCHIMEDEAN ALMOST /-R IN G S 29

5. G. Buskes and A. van Rooij, The vector lattice cover of certain partially ordered groups, J. Austral. Math. Soc. (series A), Volume 54, (1993), 352-367.

6. C.B. Huijsmans, Lattice ordered algebras and f-algebras, A Survey, in Studies in Economics Theory 2, Springer-Verlag, Berlin-Heidelberg-New York, 1990, pp. 151-169.

Gerard Buskes and Jason Holland Department of Mathematics University of Mississippi University, MS 38677 U.S.Ammbuskes@hilbert.math.olemiss.edujason.holland@oc.edu