Abstract
We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C*-algebra. We also show that if a unital C*-algebra can be approximately embedded into some tensorially self absorbing C*-algebra C (e.g., uniformly hyperfinite (UHF)-algebras of infinite type, the Cuntz algebra \({{\mathcal O}_2}\)), then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite (resp. properly infinite), the construction is also infinite (resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.
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Acknowledgements
This work was supported by the Research Center for Operator Algebras at East China Normal University which is funded by the Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400), National Natural Science Foundation of China (Grant No. 11531003), Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice (Grant No. 1361431) and the special fund for the Short-Term Training of Graduate Students from East China Normal University.
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Fu, X. Simple generalized inductive limits of C*-algebras. Sci. China Math. 64, 1029–1044 (2021). https://doi.org/10.1007/s11425-019-9513-8
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DOI: https://doi.org/10.1007/s11425-019-9513-8