On the Ideal Center of a Dual Vector Lattice

Let X be a vector lattice with order dual X′. In this paper, we investigate when Orth(X′) is an ideal center in X′. In [6], Toumi answers related question posed by Wickstead in [1]. This study is the dual version of the paper of Toumi in [6]. Mathematics Subject Classification: 46B42, 47B65


Introduction
Let  be an Archimedean vector lattice.ℎ() denotes the vector lattice of all orthomorphisms on .() denotes the sublattice of ℎ() consisting of those  for which there is non-negative real number  with − ≤ () ≤  for all  ∈  + (Positive cone of ).We say that () is the ideal center of .A vector lattice  under an associative multiplication is said to be lattice-ordered algebra (ℓ-algebra) whenever the multiplication makes  an algebra, in addition it satisfies the following property: If ,  ∈  + , then  ∈  + , [2].A lattice-ordered algebra  is said to be an -algebra when  ∧  = 0 implies  ∧  =  ∧  = 0 for each  ∈  + .Note that any Archimedean  -algebra is commutative.Moreover, squares in an -algebra are positive, [3].
If  is an -algebra, the mapping :  → ℎ() defined by () =   where   () =  for all ,  ∈  is an algebra and lattice homomorphism.If () ∈ (),  is said to be bounded and if every element of  is bounded, the -algebra  is said to be bounded, [5].
Let  be a vector lattice and let 0 ≤  ∈ .If for every real number ℰ > 0, there exists a natural number  ℰ such that |  − | ≤ ℰ for all  ≥  ℰ , the sequ-

Serap Özcan and Ömer Gök
ence {  } ≥0 in  is called () relatively uniformly convergent to  ∈  which is denoted by   → ().That is, if   → () for some 0 ≤  ∈ , then the sequence {  } ≥0 is called (relatively) uniformly convergent to .This is denoted by   → (r.u).A vector lattice is called relatively uniformly complete if every relatively uniform Cauchy sequence in  has a unique limit.Relatively uniform limits are unique if  is Archimedean.The relatively uniform topology is denoted by (r.u) topology. is called Dedekind complete, if every non-empty order bounded set has a supremum and an infimum in . is called uniformly complete if {∑    =1 ∶  ∈ } exists for every uniformly bounded sequence (  ) 1 ∞ ⊂  + , [7].
Let  be a topological algebra and () be the set of all maximal two sided ideals in .The space () is equipped with the ℎ-topology:  ⊂ () is closed if  = (()), where () is the intersection of all ideals in  and () = { ∈ ():  ⊂ } for any two-sided ideal  of  .If  is also equipped with a compatible topology we consider a subset () of () consisting of closed ideals.We use small letters ℎ and  to indicate  and  restricted to ().
Note that, for unexplained terminology and notation, we refer to the standart books [3] and [7].Next theorem is seen in the work of Basly and Triki in [5] and [6].
Theorem 1 If  is a relatively uniformly complete -algebra, the following are equivalent: . is bounded.ii.Every maximal modular ring ideal in  is uniformly closed.iii.Every maximal modular ring ideal in  is the kernel of a lattice and algebra homomorphism  → .iv.Every maximal modular ring ideal in  is a maximal order ideal.
Theorem 2 [6] Consider a relatively uniformly complete unital -algebra .Then () = (), with respect to the relatively uniform topology, if and only if i. Every proper finitely generated ring ideal is contained in a uniformly closed maximal ring ideal of .ii.() is compact in ℎ-topology.
Proposition 3 [6] Any maximal ring ideal of a relatively uniformly complete semiprime -algebra  is an order ideal.Now, we will give the dual version of the study in [6].