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**Johan OINERT ^{*} and Sergei D. SILVESTROV**

Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden E-mails: [email protected], [email protected]

- *Corresponding Author:
- Johan OINERT

Centre for Mathematical Sciences,

Lund University, Box 118,

SE-221 00 Lund, Sweden

**E-mails:**[email protected], [email protected]

**Received date: ** December 16, 2007 **Accepted Date: **April 01, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper we will give an overview of some recent results which display a connection between commutativity and the ideal structures in algebraic crossed products.

In the recent papers [3,4], we have been studying a correspondence between ideals and commutativity in algebraic crossed products. Given an algebraic crossed product , consider the following two statements:

** S1: **The coefficient ring

** S2:** For every non-zero two-sided ideal I in

In this paper we will give an overview of some types of crossed products for which the statements** S1** and

**Theorem 1.1** *([3]). Let be an algebraic crossed product and denote the commutant of A _{0} by C_{A}(A_{0}) = {a ∈ A | ab = ba, ∀b ∈ A_{0}}. If the coefficient ring A_{0} is commutative, then I ∩ C_{A}(A_{0}) ≠ {0} for every non-zero two-sided ideal I in the crossed product *

As an immediate corollary to this theorem we get that, if *A _{0}* is assumed to be maximal commutative in then

For the convenience of the reader we shall now recall the definition and the basic properties of algebraic crossed products. For more details see e.g [2]. Throughout this article all rings are assumed to be associative rings. Given a unital ring *R* we let *U(R)* denote the group of multiplication invertible elements of *R*.

**Definition 1.1.** A *G*-crossed system is a quadruple {*A _{0}*, G, σ, α}, consisting of a unital ring

*(i) σ _{x}(σy(a)) = α(x, y) σ_{xy}(a) α(x, y)<sup>−1</sup>*

*(ii) α(x, y) α(xy, z) = σ _{x}(α(y, z)) α(x, yz)*

*(iii) α(x, e) = α(e, x) = 1A _{0}*

Let *{u _{s}}S∈G* be a copy (as a set) of G. Given a

*(a _{1} u_{x})(a_{2} u_{y}) = a_{1} σ_{x}(a_{2}) α(x, y) u_{xy}*

for all a_{1}, a_{2} ∈ *A _{0}* and x, y ∈ G and extend it bilinearly to all of . Each element of G may be expressed as a formal sum for all but a finite number of g ∈ G. Explicitly, the addition and multiplication of two arbitrary elements is given by

**Proposition 1.1 **([2]). *Let {A _{0}, G, σ, α} be a G-crossed system. Then is an associative unital ring (with the multiplication defined in (1.1)).*

**Definition 1.2.** The ring is called the crossed product of the *G*-crossed system {*A _{0}*, G, σ, α}.

The coefficient ring *A _{0}* is naturally embedded as a subring into via the canonical isomorphism defined by Instead of we will simply write

**Remark 1.1.** If k is a field and A is a *k*-algebra, then so is

Depending on the nature of the maps σ and α we will give different names to the crossed product If the map α is trivial, i.e α(x, y) = 1_{A0} for every (x, y) ∈ G × G, then we shall write and refer to it as a *skew group ring*. If, on the other hand, σ is trivial, i.e. σ_{g} = id_{A0} for every g ∈ G, then we shall write and refer to it as a twisted group ring. A crossed product where both of the maps σ and α are trivial is written as and is simply refered to as a *group ring*.

If one wants to talk about maximal commutativity of *A _{0}*, it does not really make sense unless we assume that

**Example 2.1** (group rings). Let *A _{0}* be a unital ring and G any (non-trivial) group and denote the group ring by Note that this corresponds to the crossed product with trivial σ and α maps. We may define the so called augmentation map and it is straightforward to check that it is in fact a ring morphism. The kernel of this map, ker(ε) is a two-sided ideal in and it is not hard to see that ker(ε) ∩

For skew group rings we have the following theorems.

**Theorem 2.1** ([4]). If is a skew group ring where the coefficient ring *A _{0}* is an integral domain and the group G is abelian, then the two assertions

**Theorem 2.2** ([4]). If is a skew group ring where the coefficient ring *A _{0}* is commutative and G is a torsion-free abelian group, then the two assertions

**Remark 2.1.** Note that in the previous theorems, the action σ can be trivial, but in that case the situation is already described by Example 2.1.

**Example 2.2** (the algebra associated to a dynamical system). In [5,6,7] the authors studies crossed product algebras associated to dynamical systems. Suppose that we are given a nonempty set *X* and a bijection σ : X → X. Then (X, σ) is a discrete dynamical system where the action of n ∈ Z on x ∈ X is given by By C^{X} we denote the algebra of functions X → C under the usual pointwise operations of addition and multiplication. If we are given a subalgebra *A µ C ^{X}* such that it is invariant under σ and σ

In the current situation the coefficient algebra A is commutative and the group (Z, +) is clearly torsion-free and abelian, hence Theorem 2.2 is applicable. We may conclude that Theorem 2.2 is a generalization of certain parts of Corollary 4.5 in [6] and Theorem 4.5, Theorem 4.6, Corollary 4.7, Theorem 6.2 in [7].

In a twisted group ring just like for group rings mentioned above, the action σ is trivial and hence for each s ∈ G the element *u _{s}* commutes with every element in

**Example 2.3** (the field of complex numbers). Let *A _{0}* = R, G = (Z

A crossed product where neither of the maps σ and α are trivial and hence not treated in the previous section, will be refered to as a general crossed product. For this type of crossed products we are not able to say as much as we would want to.

**Theorem 3.1.** If is a crossed product where *A _{0}* is an integral domain, G is an abelian torsion-free group and α is such that α(s, t) = 1

**Proof.** It is clear from Theorem 1.1 that **S1** =) **S2**. Suppose that *A _{0}* is not maximal commutative. Since

and since *G* is abelian we get* α(gs, h) = α(sg, h) = α(g, h).* Let I be the two-sided ideal generated by 1_{A0} +*u _{s}*, which is an element that commutes with all of

Since *G* is abelian, it is clear that any element of *I* may be written in the form

for some* ct ∈ A _{0}*, where

**Remark 3.1.** Note that, a twisted group ring can never fit into the conditions of Theorem 3.1, because if σ is trivial, then the conditions force α to be trivial as well.

Finite groups are clearly not torsion-free, but Example 3.1 gives an example of a situation where **S1** and **S2** are in fact equivalent for a general crossed product graded by a finite group. This raises the question whether or not **Theorem 3.1 **can be generalized to general crossed products graded by more general groups.

**Example 3.1** (central simple algebras). Let A be a finite-dimensional central simple algebra over a field F. By Wedderburn’s theorem where D is a division algebra over *F* and *i* is some integer. If K is a maximal separable subfield of D then [K : F] = *n* where [D : F] = n^{2}. We shall assume that K is normal over F and that [A : F] = [K : F]^{2} (see [1] for motivation). Let Gal(K/F) be the Galois group of K over F. For k ∈ K and σ_{s} ∈ Gal(K/F) we shall write σ_{s}(k) for the image of k under σ_{s}. By the Noether-Skolem theorem there is an invertible element *u _{s}* ∈ A such that for every k ∈ K. One can show that the us’s are linearly independent over K. However, the linear span over

We are grateful to Freddy Van Oystaeyen for useful discussions on the topic of this article. This work was supported by the Swedish Foundation of International Cooperation in Research and Higher Education (STINT), the Crafoord Foundation, The Royal Physiographic Society in Lund, The Swedish Royal Academy of Sciences and ”LieGrits”, a Marie Curie Research Training Network funded by the European Community as project MRTN-CT 2003-505078.

- HersteinIN(1968) Noncommutative Rings.The Carus Mathematical Monographs, No. 15, The Math Assoc of America.
- NË�astË�asescuC, Van OystaeyenF (2004) Methods of Graded Rings.Lecture Notes in Math.1836,Springer-Verlag, Berlin.
- Ì�OinertJ, SilvestrovSD (2008)Commutativity and ideals in algebraic crossed products. J Gen LieTheoryAppl 2.
- Ì�OinertJ,SilvestrovSD, Ideals in crossed products and skew group rings. In preparation.
- SvenssonC, SilvestrovS, de JeuM (2007) Dynamical sytems and commutants in crossed products.Int J Math18: 455–471.
- SvenssonC, SilvestrovS, de JeuM (2006) Connections between dynamical systems and crossed products of Banach algebras byZ. In procceding”Operator Theory, Analysis and Mathematical Physics”,OTAMP-2006, June 15–22.
- SvenssonC, SilvestrovS, de JeuM, Dynamical systems associated with crossed products. In”Operator Methods in Fractal Analysis, Wavelets and Dynamical Systems PreprintarXiv:0707.1881v2 (math.OA)

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