Abstract
In this paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras.
Funding statement: This work was supported by JST CREST Grant Number JPMJCR1913 and RIKEN Special Postdoctoral Researcher Program.
Acknowledgements
The author would like to thank Prof. Takeshi Katsura for his support and encouragement. The author is also grateful to Yuhei Suzuki for fruitful discussions. In addition, the author would like to express his gratitude to the anonymous reviewer for the constructive comments and suggestions.
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