Properties expressible in small fragments of the theory of the\n hyperfinite II₁ factor

2020/07/22 by Isaac Goldbring, Goldbring, Isaac, Bradd Hart +1 · 1 citation
Computer Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2007.11628

openalex publication_date 2020/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that any II₁ factor that has the same 4-quantifier theory as the\nhyperfinite II₁ factor \R satisfies the conclusion of the Popa\nFactorial Commutant Embedding Problem (FCEP) and has the Brown property. These\nresults improve recent results proving the same conclusions under the stronger\nassumption that the factor is actually elementarily equivalent to\n\R. In the same spirit, we improve a recent result of the\nfirst-named author, who showed that if (1) the amalgamated free product of\nembeddable factors over a property (T) base is once again embeddable, and (2)\n\R is an infinitely generic embeddable factor, then the FCEP is true\nof all property (T) factors. In this paper, it is shown that item (2) can be\nweakened to assume that \R has the same 3-quantifier theory as an\ninfinitely generic embeddable factor.\n

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