Crossed Products Related to Cyclically Ordered Semi Groups

Suppose (G, R) is a cyclically ordered abelian group. Thru unwounding method of Rieger, we construct a semi group crossed product related with (G, R).


Introduction
The theory of crossed product by semigroup of endomorphisms is one of important and interesting area in modern theory of operator algebras. This theory has a plenty of applications such as in Cuntz algebras [1], Toeplitz algebras (Adji,et al. [2], Laca and Raeburn [3], Adji [4], Adji and Raeburn [5], Rosjanuardi [6], Adji and Rosjanuardi [7], Rosjanuardi and Albania [8] and Hecke algebras arising in number theory (Laca and Raeburn [9], Larsen and Raeburn [10]). This theory is a generalisation of the theory of crossed product by group of automorphism, which is a well developed area of the theory of operator algebras, and has rich of examples.
In some cases, the theory of crossed product by semigroup of endomorphisms heavily relies on theory of totally ordered group, for examples Adji et. al [2], Adji [4], Adji and Raeburn [5], Rosjanuardi [6], Adji and Rosjanuardi [7], Rosjanuardi and Albania [8], Rosjanuardi [11] are among others. The concept of order is essentials as the semigroup crossed product here is a generalisation of Stacey's [12] crossed products of single endomorphism which is related with the action of semigroup of natural numbers ℕ.
A generalisation of the notion of totally ordered group is cyclically ordered group. Cyclically ordered group which was originally introduced by Rieger in 1947 has attracted great deal of attention, for example Swierczkowski [13], Fuchs [14], Harminc [15], Jakubik [16,17], Cernak [18], Giraudet, Leloup, Lucas [19] and Leloup, Lucas [20] are among others. The concept of this order is different with regular linear order, as it is a ternary relation instead of binary relation.
Given a cyclically ordered group ( , ), we will make use the theorem of Rieger to get its unwound totally ordered group uw( ) to obtain a dynamical system and its semigroup crossed product.  Riegerin Leloup and Lucas [20] states that every cyclically ordered group can be obtained from a totally ordered group. Let ( , ≤) be a totally ordered group and 0 be a positive, central and cofinal element of . The quotient group /〈 〉 can be cyclically ordered by setting

Semigroup Crossed Products
Definition 2Let   be the positive cone of a totally ordered abelian group. A dynamical system consists of a unital C  -algebra A , and a semigroup homomorphism When we consider the semigroup , we have the dynamical system considered by Stacey [12]. Focusing on the dynamical systems considered by Stacey [12] allows us to find an example of a dynamical system that has no covariant representation. Stacey pointed out a dynamical system of no covariant representation, as we recited below.
When we consider the semigroup , we have the dynamical system considered by Stacey [12]. Focusing on the dynamical systems considered by Stacey [12] allows us to find an example of a dynamical system that has no covariant representation. Stacey pointed out a dynamical system of no covariant representation, as we recited below. Example 1 (Stacey [12]) Suppose 00 cc   is the unilateral shift, defined by The concept of semigroup crossed product of the dynamical system () A     was introduced by Adji et. al [2]. It is a development of the crossed product by groups action of Raeburn [21]. This version of crossed product can also be viewed as the untwisted version of semigroup twisted crossed product of Rosjanuardi [6,11], Adji and Rosjanuardi [7].

B is generated by ()
Existence of semigroup crossed product of a dynamical system heavily rely on the existence of non zero covariant representation. Adji,et al. [2] gave an example of dynamical system in which the crossed product exists, and is universal for isometric representation of the semi group. The algebra in the dynamical system is the closed subspace of ℓ ∞ (Γ) spanned by {1 : ∈ Γ + }, where This algebra is denoted by Γ + . For each ∈ Γ + , the automorphism of ℓ ∞ (Γ) which is defined by which implies that restricts to an action of Γ + by endomorphism of Γ + . Suppose : Γ + → (ℓ 2 (Γ + ) defined by ( ) = + , then is an isometric representation of Γ + . Proposition 2.2 of Adji, Laca, Nilsen, Raeburn [2] implies there is a representation of Γ + such that ( , ) is a covariant representation of the dynamical system ( Γ + , Γ + , ). Therefore the crossed product Γ + × Γ + exists.

Results and Discussion
Given a * −algebra , a cyclically ordered abelian group ( , ) and an action of the positive cone of . Since the class of the order is not a linear order, we can not directly make use the Definition 4 to obtain the semigroup crossed product of the dynamical system ( , , ). But we can make use the method of Rieger to unwound the cyclic order to get the linear order related with .

Example:
Given a cyclically ordered abelian group ( , ). Let uw( ) be the unwound group of ( , ). We will make use the method of Adji, Laca, Nilsen and Raeburn [2] in constructing the semi group crossed product which is universal for the isometric representation of the semi group (uw( )) + , i.e the positive cone of uw( ).
In particular, since the unwound group of ( , ) is the linearly ordered group ℤ × , then the semi group crossed product (ℤ⊕ℤ) + × (ℤ ⊕ ℤ) + considered in Rosjanuardi and Albania [8] and Rosjanuardi [11] can be viewed as a semi group crossed product related with the cyclically ordered group ℤ.

Conclusion
Given a cyclically ordered abelian group ( , ). Thru unwounding method of Rieger, we can aply the method Adji,et al [2] to obtain a semi group crossed product related with ( , ) .