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An exotic II\(_1\) factor without property Gamma

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Abstract

We introduce a new iterative amalgamated free product construction of II\(_1\) factors, and use it to construct a separable II\(_1\) factor which does not have property Gamma and is not elementarily equivalent to the free group factor \(\text {L}(\mathbb F_n)\), for any \(2\le n\le \infty \). This provides the first explicit example of two non-elementarily equivalent II\(_1\) factors without property Gamma. Moreover, our construction also provides the first explicit example of a II\(_1\) factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.

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Acknowledgements

We thank Isaac Goldbring, David Jekel, Jesse Peterson, Sorin Popa and Stefaan Vaes for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242, USA

    Ionuţ Chifan

  2. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Adrian Ioana

  3. Institute of Pure and Applied Mathematics, UCLA, 460 Portola Plaza, Los Angeles, CA, 90095, USA

    Srivatsav Kunnawalkam Elayavalli

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  1. Ionuţ Chifan
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Correspondence to Adrian Ioana.

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I.C. was partially supported by NSF FRG Grant DMS-1854194 and NSF Grant DMS-2154637; A.I. was partially supported by NSF FRG Grant #1854074, NSF Grant #DMS-2153805 and a Simons Fellowship; S.K.E was supported by a Simons Postdoctoral Fellowship.

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Chifan, I., Ioana, A. & Kunnawalkam Elayavalli, S. An exotic II\(_1\) factor without property Gamma. Geom. Funct. Anal. 33, 1243–1265 (2023). https://doi.org/10.1007/s00039-023-00649-4

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  • DOI: https://doi.org/10.1007/s00039-023-00649-4

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