An exotic II $$_1$$ factor without property Gamma

Geometric and Functional Analysis - Tập 33 - Trang 1243-1265 - 2023
Ionuţ Chifan1, Adrian Ioana2, Srivatsav Kunnawalkam Elayavalli3
1Department of Mathematics, The University of Iowa, Iowa City, USA
2Department of Mathematics, University of California San Diego, La Jolla, USA
3Institute of Pure and Applied Mathematics, UCLA, Los Angeles, USA

Tóm tắt

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.

Tài liệu tham khảo

Claire Anantharaman Delaroche and Sorin Popa, An introduction to \(\rm II_1\) factors, available at https://www.math.ucla.edu/~popa/Books/IIun.pdf.

Aim problem list, available at http://aimpl.org/groupvonneumann.

Serban Belinschi and Mireille Capitaine, Strong convergence of tensor products of independent G.U.E. matrices, available at arXiv:2205.07695.

Charles Bordenave and Benoit Collins, Norm of matrix-valued polynomials in random unitaries and permutations, available at arXiv:2304.05714v1.

Nathaniel P. Brown and Narutaka Ozawa, \({\rm C}^*\)-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, 2008.

Nathanial P. Brown, Finite free entropy and free group factors, International Mathematics Research Notices 2005 (2005), no. 28, 1709–1715.

A. Connes, Classification of injective factors. Cases \(II_{1},\)\(II_{\infty },\)\(III_{\lambda },\)\(\lambda \ne 1\), Ann. of Math. (2) 104 (1976), no. 1, 73–115.

J. Dixmier and E. C. Lance, Deux nouveaux facteurs de type \({\rm II}_{1}\), Invent. Math. 7 (1969), 226–234.

Junsheng Fang, Liming Ge, and Weihua Li, Central sequence algebras of von Neumann algebras, Taiwanese J. Math. 10 (2006), no. 1, 187–200.

Ilijas Farah, Isaac Goldbring, Dimitri Shlyakhtenko, and Wilhelm Winter, Model theory and operator algebras, available at https://www.birs.ca/workshops/2018/18w5155/report18w5155.pdf.

Junsheng Fang and Don Hadwin, A note on the invariant subspace problem relative to a type \({\rm II}_1\) factor, Houston J. Math. 37 (2011), no. 3, 879–893.

Isaac Goldbring and Bradd Hart, The universal theory of the hyperfinite \(\rm II_1\) factor is not computable, available at arXiv:2006.05629.

Isaac Goldbring and Bradd Hart, On the theories of McDuff’s \(\rm II_1\) factors, Int. Math. Res. Not. IMRN (2017), no. 18, 5609–5628.

Alin Galatan and Sorin Popa, Smooth bimodules and cohomology of \(\rm II_1\) factors, J. Inst. Math. Jussieu 16 (2017), no. 1, 155–187.

Ben Hayes, A random matrix approach to the Peterson-Thom conjecture, to appear in Indiana Univ. Math. Journal (2020).

C. W. Henson and J. Iovino, Ultraproducts in analysis, Analysis and logic (Mons, 1997), London Math. Soc. Lecture Note Ser., vol. 262, Cambridge Univ. Press, Cambridge, 2002, pp. 1–110.

Ben Hayes, David Jekel, and Srivatsav Kunnawalkam Elayavalli, Peterson-Thom conjecture and strong 1-boundedness for von neumann algebras, In preparation, 2022.

Adrian Ioana and Jesse Peterson, Classification problems in von neumann algebras, available at https://www.birs.ca/workshops/2019/19w5134/report19w5134.pdf.

David Jekel, Covering entropy for types in tracial \({\rm W}^*\)-algebras, 2022.

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, \(\text{MIP}^*\)=RE, available at arXiv:2001.04383.

Dusa McDuff, Uncountably many \({\rm II}_{1}\) factors, Ann. of Math. (2) 90 (1969), 372–377.

Jesse Peterson, Open problems in operator algebras, available at https://math.vanderbilt.edu/peters10/problems.html.

Sorin Popa, On derivations into the compacts and some properties of type \({\rm II}_1\) factors, Spectral theory of linear operators and related topics (Timişoara/Herculane, 1983), Oper. Theory Adv. Appl., vol. 14, Birkhäuser, Basel, 1984, pp. 221–227.

Sorin Popa, Correspondences, INCREST preprint, unpublished. (1986).

Sorin Popa, On a class of type \({\rm II}_1\) factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), no. 3, 809–899.

Sorin Popa and Stefaan Vaes, Group measure space decomposition of \({\rm II}_1\) factors and \(W^\ast \)-superrigidity, Invent. Math. 182 (2010), no. 2, 371–417.

Hui Tan, Spectral gap characterizations of property (T) for \(\rm II_1\) factors, available at arXiv:2202.06089.

G. Zeller-Meier, Deux nouveaux facteurs de type \({\rm II}_{1}\), Invent. Math. 7 (1969), 235–242.