ON THE TOPOLOGY OF THE DUAL SPACE OF CROSSED PRODUCT C*-ALGEBRAS WITH FINITE GROUPS

• Kamalov, Firuz (Mathematics Department Canadian University of Dubai)
• Published : 2017.03.31

Abstract

In this note we extend our previous result about the structure of the dual of a crossed product $C^*$-algebra $A{\rtimes}_{\sigma}G$, when G is a finite group. We consider the space $\tilde{\Gamma}$ which consists of pairs of irreducible representations of A and irreducible projective representations of subgroups of G. Our goal is to endow $\tilde{\Gamma}$ with a topology so that the orbit space e $G{\backslash}{\tilde{\Gamma}}$ is homeomorphic to the dual of $A{\rtimes}_{\sigma}G$. In particular, we will show that if $\widehat{A}$ is Hausdorff then $G{\backslash}{\tilde{\Gamma}}$ is homeomorphic to $\widehat{A{\rtimes}_{\sigma}G}$.

References

1. A. Arias and F. Latremoliere, Irreducible representations of C*-crossed products by finite groups, J. Ramanujan Math. Soc. 25 (2010), no. 2, 193-231.
2. S. Echterhoff and H. Emerson, Structure and K-theory for crossed products by proper actions, Expo. Math. 29 (2011), no. 3, 300-344. https://doi.org/10.1016/j.exmath.2011.05.001
3. S. Echterhoff and D. Williams, Structure of crossed products by strictly proper actions on continuous trace algebras, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3649-3673. https://doi.org/10.1090/S0002-9947-2014-06263-6
4. F. Kamalov, The dual structure of crossed product C*-algebras with finite groups, Bull. Aust. Math. Soc. 88 (2013), no. 2, 243-249. https://doi.org/10.1017/S0004972712001049
5. M. Rieffel, Actions of finite groups on C*-algebras, Math. Scand. 47 (1980), no. 1, 157-176. https://doi.org/10.7146/math.scand.a-11882
6. M. Takesaki, Covariant representations of C*-algebras and their locally compact auto-morphism groups, Acta Math. 119 (1967), 273-303. https://doi.org/10.1007/BF02392085
7. D. Williams, The topology on the primitive ideal space of transformation group C*-algebras and CCR transformation group C*-algebras, Trans. Amer. Math. Soc. 266 (1981), no. 2, 335-359. https://doi.org/10.1090/S0002-9947-1981-0617538-7
8. D. Williams, Crossed products of C*-algebras, Mathematical Surveys and Monographs, Vol. 134, American Mathematical Society, Providence, RI, 2007.