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Tracially sequentially split \(^*\)-homomorphisms between \(C^*\)-algebras

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Abstract

We define a tracial analog of the notion called a sequentially split \(^*\)-homomorphism between \(C^*\)-algebras due to Barlak and Szabó and show that several important approximation properties related to the classification theory of \(C^*\)-algebras pass from the target algebra to the domain algebra. Then, we show that this framework arises from tracial Rokhlin properties of a finite group action and an inclusion of unital \(C^*\)-algebras.

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Acknowledgements

A main part of this research was carried out during the first author’s stay at KIAS and his visit to Ritsumeikan University from 2017 to 2018. He would like to appreciate both institutions for their excellent support. Both authors are grateful to a colleague for pointing out a flaw in our original definition about “tracially sequentially-split \(^*\)-homomorphism” and Chris Phillips for suggesting an outline to fill a gap in the proof of Theorem 2.22. In addition, we are grateful to T. Teruya for a helpful discussion. Finally, we deeply thank the referee for suggesting the idea of using the ultrapower algebra. Hyun Ho Lee’s research was supported by 2018 Research Fund of University of Ulsan. Hiroyuki Osaka’s research was partially supported by the JSPS Grant for Scientific Research nos. 17K05285 and 20K03644.

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Authors and Affiliations

  1. Department of Mathematics, University of Ulsan, Ulsan, 44610, Republic of Korea

    Hyun Ho Lee

  2. Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577, Japan

    Hiroyuki Osaka

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  1. Hyun Ho Lee
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  2. Hiroyuki Osaka
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Correspondence to Hyun Ho Lee.

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Communicated by Baruch Solel.

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Lee, H.H., Osaka, H. Tracially sequentially split \(^*\)-homomorphisms between \(C^*\)-algebras. Ann. Funct. Anal. 12, 37 (2021). https://doi.org/10.1007/s43034-021-00115-y

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  • DOI: https://doi.org/10.1007/s43034-021-00115-y

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