The bidual of r-algebras
We prove that the order continuous bidual of an Archimedean r-algebra is a Dedekind complete r-algebra with respect to the Arens multiplications.
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| author | Yilmaz, R. Ілмаз, Р. |
| author_facet | Yilmaz, R. Ілмаз, Р. |
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| description | We prove that the order continuous bidual of an Archimedean r-algebra is a Dedekind complete r-algebra with respect to the Arens multiplications. |
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UDC 512.5
R. Yilmaz (Rize Univ., Turkey)
THE BIDUAL OF r-ALGEBRAS
БIДУАЛ r-АЛГЕБР
We prove that the order continuous bidual of an Archimedean r-algebra is a Dedekind complete r-algebra
with respect to the Arens multiplications.
Доведено, що порядковий неперервний бiдуал архiмедової r-алгебри є повною r-алгеброю Дедекiнда
вiдносно множень Аренса.
1. Introduction. In [11] we studied a new class of lattice ordered algebras; so-called r-
algebras and presented its relation with the certain lattice ordered algebras; f -algebras [5]
(a lattice ordered algebra A with the property that a∧ b = 0 implies ac∧ b = ca∧ b = 0
for all c ∈ A+), almost f -algebras [6] (a lattice ordered algebra A for which a ∧ b = 0
in A implies ab = 0), d-algebras [9] (a lattice ordered algebra A such that a ∧ b = 0
in A implies ac ∧ bc = ca ∧ cb = 0 for all c ∈ A+), pseudo f -algebras [7] (a lattice
ordered algebra A having the property that ab = 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 = 0). A lattice ordered algebra A in which a ∧ b = 0 in A
implies ab ∧ ba = 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. Hence an r-algebra is a generalization of a d-algebra in much the same
way as an almost f -algebra is a generalization of an f -algebra. Observe also that the
Archimedean r-algebra A is not commutative (for details, see [11]).
In this paper we concentrate on the Arens multiplications [2, 3] in the algebraic
bidual of r-algebras and prove that the order continuous bidual of an Archimedean r-
algebra is again a Dedekind complete (and hence Archimedean) r-algebra. This is the
extension of a result of Bernau and Huijsmans in [4] in which they prove that the order
continuous bidual of an almost f -algebra (respectively d-algebra) is again an almost
f -algebra (respectively d-algebra).
We now assume that A is a lattice ordered algebra, which is not necessarily commu-
tative or unital. The following two multiplications in A′′ can be introduced, which are
referred to as the first and second Arens multiplications [2, 3]. They are accomplished
in three steps: for a, b ∈ A, f ∈ A′ and F,G ∈ A′′, define a · f, f · F : A 7→ R and
F ·G : A′ 7→ R (fa, Ff and FG for the second multiplication) by
(a · f)(b) = f(ba),
(f · F )(a) = F (a · f),
(F ·G)(f) = G(f · F )
and
(fa)(b) = f(ab),
(Ff)(a) = F (fa),
(FG)(f) = F (Gf).
c© R. YILMAZ, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5 713
714 R. YILMAZ
We shall concentrate on the first Arens multiplication; similar results hold for the second.
For the elementary theory of `-spaces and terminology not explained here we refer
to [1, 10, 12].
2. The order continuous bidual of r-algebras. In this section we consider the
order continuous bidual of the class of Archimedean r-algebras and prove that the
order continuous bidual of an r-algebra A is again an r-algebra with respect to the
Arens multiplication. We first recall some relevant notions. The canonical mapping
a 7→ â of a vector lattice A into its order bidual A′′ is defined by â(f) = f(a) for all
f ∈ A′. For each a ∈ A, â defines an order continuous algebraic lattice homomorphism
on A′ and the canonical image  of A is a subalgebra of (A′)′c. Moreover the band
S = {F ∈ (A′)′c : |F | ≤ x̂ for some x ∈ A+} generated by  is order dense in (A′)′c;
that is, for each F ∈ (A′)′c, there exists an upwards directed net {Gλ : λ ∈ Λ} in SÂ
such that 0 < Gλ ↑ F.
Lemma 2.1. Let A be an r-algebra and 0 ≤ G,H ∈ (A′)′c. If G ∧ H = 0 and
G,H ≤ x̂ for some x ∈ A+, then G ·H ∧H ·G = 0.
Proof. Let 0 ≤ f ∈ A′ and x ∈ A+. Then define the positive linear functional fx
in A′ by (fx)(y) = f(xy) for all y ∈ A. It follows that f · x̂ = fx since
(f · x̂)(y) = x̂(y · f) = (y · f)(x) = f(xy) = (fx)(y)
for all y ∈ A. Hence 0 ≤ x · f + fx ∈ A′, and so, by Corollary 1.2 of [4], there exist
g, h ∈ A′ with g ∧ h = 0, and G(g) = 0 = H(h) such that x · f + fx = g + h. Hence
inf{g(y) + h(z) : x = y + z, y, z ∈ A+} = (g ∧ h)(x) = 0,
which implies that, for ε > 0, there exist y, z ∈ A+ such that x = y + z and g(y) < ε
and h(z) < ε.
We now define the linear functionals G1 and H1 on A′ by
G1 = G ∧ ̂(y − y ∧ z) and H1 = H ∧ ̂(z − y ∧ z).
Clearly, 0 ≤ G1, H1 ∈ (A′)′c and the following inequalities hold:
0 ≤ H −H1 = (H − (z − y ∧ z))+ ≤ (x̂− ̂(z − y ∧ z))+
= ( ̂y + z − (z − y ∧ z))+ = ( ̂y + y ∧ z)+ ≤ 2ŷ, (1)
and similarly
0 ≤ G−G1 ≤ 2ẑ. (2)
Since (y − y ∧ z) ∧ (z − y ∧ z) = (y ∧ z)− (y ∧ z) = 0 in A and A is an r-algebra,
(y − y ∧ z)(z − y ∧ z) ∧ (z − y ∧ z)(y − y ∧ z) = 0,
and so 0 ≤ G1 ·H1∧H1 ·G1 ≤ ̂(y − y ∧ z) · ̂(z − y ∧ z)∧ ̂(z − y ∧ z) · ̂(y − y ∧ z) = 0;
i.e.,
G1 ·H1 ∧H1 ·G1 = 0. (3)
We next consider the elements 0 ≤ G · (H −H1), (G − G1) ·H1, H · (G − G1) and
(H −H1) ·G1 of (A′)′c. Then, by (1),
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
THE BIDUAL OF r-ALGEBRAS 715
(G · (H −H1))(f) ≤ (x̂ · (H −H1))(f) = (H −H1)(f · x̂) =
= (H −H1)(fx) ≤ (H −H1)(fx+ x · f) =
= (H −H1)(g + h) = (H −H1)(g) + (H −H1)(h) ≤
≤ (H −H1)(g) +H(h) ≤ 2ŷ(g) + 0 = 2g(y) (4)
and, by (2),
((G−G1) ·H1)(f) ≤ ((G−G1) · x̂)(f) = x̂(f · (G−G1)) =
= (f · (G−G1))(x) = (G−G1)(x · f) ≤
≤ (G−G1)(x · f + fx) = (G−G1)(g + h) =
= (G−G1)(g) + (G−G1)(h) ≤
≤ G(g) + (G−G1)(h) ≤ 2h(z). (5)
It follows by symmetry that
(H · (G−G1))(f) ≤ 2h(z) and ((H −H1) ·G1)(f) ≤ 2g(y). (6)
Using the fact that (a+ b) ∧ c ≤ a ∧ c+ b ∧ c ≤ a+ b ∧ c in `-spaces and (3), we find
G ·H ∧H ·G = [
(
G · (H −H1) + (G−G1) ·H1 +G1 ·H1
)
]∧
∧[H · (G−G1) + (H −H1) ·G1 +H1 ·G1] ≤
≤ G · (H −H1) + (G−G1) ·H1+
+G1 ·H1 ∧
[
H · (G−G1) + (H −H1) ·G1 +H1 ·G1
]
≤
≤ G · (H −H1) + (G−G1) ·H1+
+G1 ·H1 ∧ (H · (G−G1) + (H −H1) ·G1)+
+G1 ·H1 ∧H1 ·G1 ≤
≤ G · (H −H1) + (G−G1) ·H1+
+H · (G−G1) + (H −H1) ·G1.
Hence, by (4), (5) and (6),
0 ≤ (G ·H ∧H ·G)(f) ≤ ((G · (H −H1))(f) + ((G−G1) ·H1))(f)+
+(H · (G−G1))(f) + ((H −H1) ·G1))(f) ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
716 R. YILMAZ
≤ 2g(y) + 2h(z) + 2h(z) + 2g(y) ≤ 8ε.
Since this holds for an arbitrary ε > 0, we have (G · H ∧ H · G)(f) = 0 for all
0 ≤ f ∈ A′. It now follows that for all f ∈ A′
(G ·H ∧H ·G)(f) = (G ·H ∧H ·G)(f+)− (G ·H ∧H ·G)(f−) = 0,
and so G ·H ∧H ·G = 0.
We now in a position to express the main result of this work.
Theorem 2.1. The order continuous bidual of an r-algebra is a Dedekind com-
plete (and hence Archimedean) r-algebra.
Proof. Let A be an r-algebra. We only need to prove G ·H ∧H ·G = 0 whenever
0 ≤ G,H ∈ (A′)′c with G ∧ H = 0. To this end, consider the band SÂ generated by
the canonical image  of A in (A′)′c. Since S is order dense in (A′)′c, there exist
Gα, Hβ ∈ SÂ such that 0 ≤ Gα ↑ G and 0 ≤ Hβ ↑ H with 0 ≤ Gα ≤ x̂α and
0 ≤ Hβ ≤ ŷβ for some xα, yβ ∈ A+. It follows from G∧H = 0 that Gα ∧Hβ = 0 for
all α, β. Furthermore, 0 ≤ Gα, Hβ ≤ x̂α + yβ . Hence, by Lemma 2.1, we see that
Gα ·Hβ ∧Hβ ·Gα = 0 (7)
for all α and β. Now let 0 ≤ f ∈ A′. It follows from 0 ≤ Gα ↑ G that 0 ≤
≤ Gα(x · f) ↑ G(x · f); i.e., 0 ≤ (f · Gα)(x) ↑ (f · G)(x) for all x ∈ A+. This
shows that 0 ≤ f · Gα ↑ f · G. Hence, by the order continuity of Hβ for each β,
0 ≤ Hβ(f ·Gα) ↑ Hβ(f ·G); i.e., 0 ≤ (Gα ·Hβ)(f) ↑ (G ·Hβ)(f), which implies that,
for each β ,
0 ≤ Gα ·Hβ ↑ G ·Hβ . (8)
Similarly, since 0 ≤ Hβ ↑ H, 0 ≤ Hβ(f · G) ↑ H(f · G); i.e., 0 ≤
≤ (G ·Hβ)(f) ↑ (G ·H)(f) for all 0 ≤ f ∈ A′, and so
0 ≤ G ·Hβ ↑ G ·H. (9)
In the same way, using the order continuity of Gα for each α, we obtain
0 ≤ Hβ ·Gα ↑ H ·Gα, (10)
leading to
0 ≤ H ·Gα ↑ H ·G. (11)
It follows from (8) and (10) that 0 ≤ Gα ·Hβ∧Hβ ·Gα ↑ G·Hβ∧H ·Gα, and so, by (7),
G·Hβ∧H ·Gα = 0 for all α and β. On the other hand, 0 ≤ G·Hβ∧H ·Gα ↑ G·H∧H ·G
by (9) and (11), and so G ·H ∧H ·G = 0, as required.
As remarked earlier, the order bidual A′′ of an almost f -algebra (respectively f -
algebra) A is an almost f -algebra (respectively f -algebra) which may not be true for
the order biduals of either d-algebras [4] or r-algebras. However we have the following
consequence.
Corollary 2.1. The order bidual of a commutative r-algebra is a Dedekind com-
plete r-algebra.
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2. Arens R. Operations induced in function classes // Monatsh. Math. – 1951. – 55. – P. 1 – 19.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
THE BIDUAL OF r-ALGEBRAS 717
3. Arens R. The adjoint of a bilinear operation // Proc. Amer. Math. Soc. – 1951. – 2. – P. 839 – 848.
4. Bernau S. J., Huijsmans C. B. The order bidual of almost f -algebras and d-algebras // Trans. Amer.
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Received 25.10.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 5
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| id | umjimathkievua-article-2756 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:29:44Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-27562020-03-18T19:35:13Z The bidual of r-algebras Бiдуал r-алгебр Yilmaz, R. Ілмаз, Р. We prove that the order continuous bidual of an Archimedean r-algebra is a Dedekind complete r-algebra with respect to the Arens multiplications. Доведено, що порядковий неперервний бiдуал архiмедової r-алгебри є повною r-алгеброю Дедекiнда вiдносно множень Аренса. Institute of Mathematics, NAS of Ukraine 2011-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2756 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 5 (2011); 713-717 Український математичний журнал; Том 63 № 5 (2011); 713-717 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2756/2268 https://umj.imath.kiev.ua/index.php/umj/article/view/2756/2269 Copyright (c) 2011 Yilmaz R. |
| spellingShingle | Yilmaz, R. Ілмаз, Р. The bidual of r-algebras |
| title | The bidual of r-algebras |
| title_alt | Бiдуал r-алгебр |
| title_full | The bidual of r-algebras |
| title_fullStr | The bidual of r-algebras |
| title_full_unstemmed | The bidual of r-algebras |
| title_short | The bidual of r-algebras |
| title_sort | bidual of r-algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2756 |
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