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 .