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Relative commutants of strongly self-absorbing \(\mathrm {C}^*\)-algebras

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Abstract

The relative commutant \(A^{\prime }\cap A^{\mathcal U}\) of a strongly self-absorbing algebra A is indistinguishable from its ultrapower \(A^{\mathcal U}\). This applies both to the case when A is the hyperfinite II\(_1\) factor and to the case when it is a strongly self-absorbing \(\mathrm {C}^*\)-algebra. In the latter case, we prove analogous results for \(\ell _\infty (A)/c_0(A)\) and reduced powers corresponding to other filters on \({\mathbb N}\). Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.

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Notes

  1. All ultrafilters are assumed to be nonprincipal ultrafilters on \({\mathbb N}\) and \(A^{\mathcal U}\) denotes the ultrapower of A associated with \(\mathcal U\).

  2. \(\otimes _{\mathrm {min}}\) denotes the minimal, or spatial, tensor product of \(\mathrm {C}^*\)-algebras. In most of this paper, we work with tensor products where one factor is nuclear, and then simply write \(\otimes \) (since all \(\mathrm {C}^*\)-tensor norms are the same).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, ON, M3J 1P3, Canada

    Ilijas Farah

  2. Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia

    Ilijas Farah

  3. Department of Mathematics and Statistics, McMaster University, 1280 Main Street, West Hamilton, ON, L8S 4K1, Canada

    Bradd Hart

  4. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Mikael Rørdam

  5. Institute of Mathematics, University of Aberdeen, Aberdeen, UK

    Aaron Tikuisis

Authors
  1. Ilijas Farah
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  2. Bradd Hart
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  3. Mikael Rørdam
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  4. Aaron Tikuisis
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Corresponding author

Correspondence to Bradd Hart.

Additional information

AT was supported by an NSERC PDF. MR was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at University of Copenhagen, and The Danish Council for Independent Research, Natural Sciences. IF and BH were partially supported by NSERC.

We are indebted to Leonel Robert for proving (2) of Theorem 1 and giving a large number of very useful remarks.

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Farah, I., Hart, B., Rørdam, M. et al. Relative commutants of strongly self-absorbing \(\mathrm {C}^*\)-algebras. Sel. Math. New Ser. 23, 363–387 (2017). https://doi.org/10.1007/s00029-016-0237-y

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