Simplicity of Skew Group Rings of Abelian Groups

Abstract

Necessary and sufficient conditions for simplicity of a general skew group ring A ⋊_σ G are not known. In this article, we show that a skew group ring A ⋊_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ⋊_σ G is simple.

Key Words:

• Ideals
• Minimality
• Skew group rings
• Simplicity
• Transformation groups

2010 Mathematical Subject Classification:

• 16S35
• 16D25
• 37B05

ACKNOWLEDGMENTS

The author is immensely grateful to Steven Deprez for stimulating discussions and in particular for sharing his knowledge on ICC groups, which gave rise to Section 5.1. The author is also grateful to Patrik Lundström for stimulating discussions on the topic of this article. This research was supported by The Swedish Research Council (postdoctoral fellowship no. 2010-918) and The Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation.

The author recently became aware of that A. D. Bell (1987), J. Algebra 105(1):76–115, has shown an analogue to Theorem 1.2(c) for any ring which is strongly graded by a hypercentral group. This result has been generalized by E. Jespers (1993), Comm. Algebra 21(7):2437–2444, who has shown that one need not assume that the gradation is strong. However, the method of proof that we use in the current paper is more direct than the ones used by Bell and Jespers.

Notes

¹The completion of this skew group algebra with respect to a suitable norm would be called a crossed product C*-algebra by C*-algebraists. In noncommutative ring theory, however, a skew group algebra is a special case of the more general (algebraic) crossed product construction.

Communicated by M. Cohen.