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On Units and Identities in Topological Lattice Ordered Algebras International Mathematical Forum, Vol. 6, 2011, no. 21, 1017 - 1021 On Units and Identities in Topological Lattice Ordered Algebras Rusen Yilmaz Department of Mathematics Faculty of Arts and Science Rize University 53100 Rize, Turkey [email protected] Abstract In this short paper we investigate characterizations of various type of order units and approximate identities in topological lattice ordered algebras. Mathematics Subject Classification: 06F25, 46A42 Keywords: Order unit, approximate identity, vector lattice, Riesz space, normed lattice ordered algebra, normed Riesz algebra, Banach lattice algebra For the elementary theory of vector lattices and terminology not explained here we refer to [1, 6, 9]. A positive element u in a vector lattice (Riesz space) A is an order unit whenever for each a ∈ A there exists some λ>0 with |a|≤λu, and a positive element u in a normed vector lattice A is a norm order unit whenever a ∈ A and a≤1 imply that |a|≤u. A positive element u in a vector lattice A is a weak order unit whenever a ∧ nu ↑ a holds for each a ∈ A+. In a similar manner, we can define topological order units in a normed vector lattice A, which are also known as quasi-interior points of A. A positive element u in a normed vector lattice A is said to be a topological order unit whenever + limn→∞ a − a ∧ nu = 0 for each a ∈ A . Recall that if an ↑ in a normed vector lattice A and limn→∞ a − an = 0, then a = sup an (see, for example, Theorems 15.3 of [9]). So, we list some consequences below. (1) Every norm order unit in a normed vector lattice is a topological order unit. (2) Every topological order unit in a normed vector lattice is a weak order unit. 1018 Rusen Yilmaz (3) If u is a positive element of a normed vector lattice and Au is the closure of the ideal Au generated by u, then u is a topological order unit in Au. (4) In a Banach lattice with order continuous norm every weak order unit is a topological order unit; that is, for such Banach lattices weak order units and topological order units turn out to be the same. (The norm in a Banach lattice (A, ·) is said to be order continuous if and only if aα ↓ 0inA implies aα→0.) Not all weak order units are topological order units. For example, let u ∈ C([0, 1]) be such that u(x)=x for all 0 ≤ x ≤ 1. Then u is a weak order unit, and 1 − 1 ∧ nu = 1. Hence u is not a topological order unit. Summarizing our results for the case of algebra gives the following relations and characterizations. In a normed lattice ordered algebras norm order unit ⇒ topological order unit ⇒ weak order unit, in a Banach lattice algebra with a multiplicative identity e ≥ 0 e is a (norm) order unit ⇔ it is a topological order unit ⇒ it is a weak order unit and in a Banach lattice algebra with a multiplicative identity e ≥ 0 and order continuous norm e is a topological order unit ⇔ it is a weak order unit. Definition 1 Let A be a lattice ordered algebra (Riesz algebra). An order + bounded net {eλ : λ ∈ Λ}↑in A is said to be + (i) an approximate identity if aeλ ↑ a and eλa ↑ a for all a ∈ A . (ii) a weak approximate identity (σ(A, A )-approximate identity) if f(aeλ) ↑ + + f(a) and f(eλa) ↑ f(a) for all a ∈ A and all f ∈ (A ) . Obviously, every multiplicative identity in a lattice ordered algebra is a weak approximate identity and an approximate identity. Definition 2 Let A be a normed lattice ordered algebra. An order bounded + net {eλ : λ ∈ Λ}↑in A is said to be (i) a norm approximate identity if aeλ − a→0 and eλa − a→0 for all a ∈ A+. (ii) a weak norm approximate identity (σ(A, A)-norm approximate identity) + if |f(aeλ) − f(a)|→0 and |f(eλa) − f(a)|→0 for all a ∈ A and all f ∈ (A)+. On units and identities in topological lattice ordered algebras 1019 We note that every multiplicative identity in a normed lattice ordered al- gebra is a norm approximate identity. Moreover, we have the following. Lemma 3 In a normed lattice algebra, the following hold. norm approximate identity ⇒ weak norm approximate identity ⇒ weak approximate identity ⇒ approximate identity. Proof. Clearly, every weak norm approximate identity is a weak approximate identity. Every norm approximate identity in A is a weak norm approximate identity; + for, if an upwards directed net {eλ : λ ∈ Λ} in A is a norm approximate identity in A, then |f(aeλ) − f(a)| = |f(aeλ − a)|≤faeλ − a→0, and |f(eλa) − f(a)|≤faeλ − a→0, as required. + Suppose that {eλ : λ ∈ Λ}↑in A is a weak norm approximate identity. + Then |f(aeλ) − f(a)|→0 and |f(eλa) − f(a)|→0 for all a ∈ A and all + + f ∈ (A ) , which imply that f(aeτ ) ↑ f(a) and f(eτ a) ↑ f(a) for all a ∈ A and all f ∈ (A)+, as required. Every weak approximate identity in a lattice ordered algebra is an approx- imate identity, as follows. + If a net {eλ : λ ∈ Λ}↑in A is a weak approximate identity, then + + supλ f(aeλ) = supλ f(eλa)=f(a) for all f ∈ (A ) and a ∈ A . Suppose + that supλ(aeλ) <aand that supλ(aeλ)=b for some b ∈ A such that b = a. Clearly, a>a∧b, and so, a−a∧b>0. Thus f(a−a∧b) > 0 for some f ∈ (A)+ since A separates the points of A. Hence f(a) >f(a∧b). On the other hands, since and aeλ ≤ b, and so aeλ ≤ a ∧ b, we have that f(a) >f(a ∧ b) ≥ f(aeλ) for all λ. Hence, f(a) > supλ f(aeλ), which is a contradiction to the hypoth- + esis. Therefore we see that supλ(aeλ)=a for all a ∈ A . Similarly we have + supλ(eλa)=a for all a ∈ A , as required. In the case of Banach lattices having order continuous norm, the reverse implications above also hold. Theorem 4 Let A be a Banach lattice algebra with an order continuous norm. + For an order bounded net {eλ : λ ∈ Λ}↑in A , the following statements are equivalent. (1) It is a norm approximate identity. (2) It is a weak norm approximate identity. (3) It is a weak approximate identity . (4) It is an approximate identity. 1020 Rusen Yilmaz Proof. By Lemma 3, it is sufficient to show that (4) ⇒ (1). Assume that {eλ : + λ ∈ Λ}↑in A is an approximate identity. Then supλ(aeλ) = supλ(eλa)=a; + that is, a − aeλ = a − eλa ↓ 0 for all a ∈ A . It follows that a − aeλ→ + 0 and a − eλa→0 for all a ∈ A since A is a Banach lattice with an order continuous norm (see, for example, [1, Theorem 12.9]). Remark 5 In general, if {xλ : λ ∈ Λ} is a net in any Banach lattice such that supλ xλ = x exists, then it is not necessarily true that x = limλ xλ.For example, consider the space ∞ of all bounded sequences of real numbers, with the usual operations of addition, scalar multiplication, partial ordering and the norm x = sup |xn|, x ∈ ∞.(∞, ·) is a Dedekind complete Banach lattice. Let xn = {1,...,1, 0,...}∈∞ (n =1, 2,...), where the 1’s occupy the first n positions. Then sup xn = {1, 1,...}∈∞ but {xn} is not norm Cauchy since xn − xm = 1 for n = m. We observe here that the norm · on ∞ is not order continuous (see, for example, [1, Theorem 14.4]). Therefore the order continuity property in the above result is essential. We complete the paper with a remark on a characterization of certain classes of lattice ordered algebras with multiplicative identity. Definition 6 A lattice ordered algebra A (or, a Riesz algebra; that is, A is a linear algebra such that ab ∈ A+ for all a, b ∈ A+) said to be (1) a f-algebra if a ∧ b =0inA implies ac ∧ b = ca ∧ b = 0 for all c ∈ A+. (2) a d-algebra if a∧b =0inA implies ac∧bc = ca∧cb = 0 for all c ∈ A+. (3) an almost f-algebra if a ∧ b =0inA implies ab =0. The notion of an f-algebra, as given in the above definition, first appeared in a paper by Birkhoff and Pierce [4] in 1956. The class of d-algebras was introduced by Kudl´aˇcek [5, 1962]. The notion of an almost f-algebra is due to Birkhoff [3, 1967]. Although these various classes of lattice ordered algebras are distinct, there are relations between them; for example, every f-algebra is an almost f-algebra and a d-algebra, and the classes of almost f-algebras and d-algebras are r- algebras (for characterizations and properties of these classes of lattice ordered algebras, see [2] or [8]). Theorem 7 In an Archimedean (associative) lattice ordered algebras A with a multiplicative identity e ≥ 0, the following statements are equivalent (e =0 whenever A is trivial, i.e., A = {0}). (1) A is an f-algebra. (2) A is an almost f-algebra. (3) A is a d-algebra. (4) e is a weak order unit. On units and identities in topological lattice ordered algebras 1021 If A is a normed f-algebra with a multiplicative identity e ≥ 0, then e is a norm order unit [7, 2.1. Lemma]. It is also proved in [7, 2.2 Proposition] that if A is a Banach lattice algebra with a multiplicative identity e ≥ 0 which is also a topological order unit of A, then e is a norm order unit.
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