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3 results

  • THE UNIVERSAL THEORY OF THE HYPERFINITE II$_1$ FACTOR IS NOT COMPUTABLE

  • ISAAC GOLDBRING, BRADD HART
  • Journal:
    Bulletin of Symbolic Logic / Volume 30 / Issue 2 / June 2024
    Published online by Cambridge University Press:
    16 February 2024, pp. 181-198
    Print publication:
    June 2024
    • Article
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  • We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg’s QWEP Conjecture and Tsirelson’s Problem.

  • 13 - Kirchberg’s conjecture

  • Gilles Pisier, Texas A & M University
  • Book:
    Tensor Products of <I>C</I>*-Algebras and Operator Spaces
    Published online:
    10 February 2020
    Print publication:
    27 February 2020, pp 280-290

  • Summary

    Here we formulate the Connes embedding problem, whether any tracial probability space embeds in an ultraproduct of matricial ones. We also briefly describe the so-called hyperfinite factor R, with which one can reformulate the question as asking for an embedding in an ultrapower of R. Since the Connes problem is open even for the tracial probability spaces associated to discrete groups, this leads us to describe several related interesting classes of infinite groupssuch as residually finite, hyperlinear and sofic groups. We also discuss the so-called matrix models in terms of which the Connes problem can be naturally reformulated. Lastly, we give a quite transparent characterization of nuclear von Neumann algebras, which shows that there are very few of them.