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Trivial endomorphisms of the Calkin algebra

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Abstract

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra \({\cal Q}(H)\) is unitarily equivalent to an endomorphism of \({\cal Q}(H)\) which is liftable to a unital endomorphism of \({\cal B}(H)\). We use this result to classify all unital endomorphisms of \({\cal Q}(H)\) up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of C*-algebras that embed into \({\cal Q}(H)\) is not closed under tensor product nor countable inductive limit.

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Acknowledgements

I wish to thank Ilijas Farah for his suggestions concerning these problems and for his useful remarks on the early drafts of this paper. I also would like to thank Alessandro Vignati for the valuable conversations we had about these topics. Finally, I thank the anonymous referee for their helpful remarks on the paper.

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  1. Department of Mathematics, Université de Paris, 1 Place Aurélie Nemours, 75013, Paris, France

    Andrea Vaccaro

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  1. Andrea Vaccaro
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Correspondence to Andrea Vaccaro.

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Vaccaro, A. Trivial endomorphisms of the Calkin algebra. Isr. J. Math. 247, 873–903 (2022). https://doi.org/10.1007/s11856-021-2284-0

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  • DOI: https://doi.org/10.1007/s11856-021-2284-0

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