1. Principles of operator algebras
  2. Spectral measures
  3. 16d. Big index

16d. Big index

[math] \newcommand{\mathds}{\mathbb}[/math]

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In big index now, the philosophy is that the index of subfactors [math]N\in[1,\infty)[/math] should be regarded as being the well-known [math]N[/math] variable from physics, which must be big:

[[math]] N\to\infty [[/math]]


More precisely, the idea is that the constructions involving groups, group duals, or more generally compact quantum groups, producing subfactors of integer index, [math]N\in\mathbb N[/math], can be used with “uniform objects” as input, and so produce an asymptotic theory.


The problem however is how to axiomatize the uniformity notion which is needed, in order to have some control on the resulting planar algebra [math]P=(P_k)[/math]. The answer here comes from the notion of easiness, that we already met in chapter 8, and its various technical extensions, which are in fact not currently unified, or even fully axiomatized.


The main technical questions here are the classification of the easy quantum groups on one hand, and the axiomatization of the super-quizzy quantum groups on the other hand. We also have the question of better understanding the relation between easiness, subfactors, planar algebras, noncommutative geometry and free probability, and we refer here to [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].


Summarizing, we have many interesting questions, both in small and big index. As a common ground here, both these questions happen inside the Murray-von Neumann factor [math]R[/math], although this is conjectural in big index, related to existence questions for outer actions and matrix models. Thus, as a good problem to finish with, which is from Jones' original subfactor paper [11], and is due to Connes, we have the question of axiomatizing the finite index subfactors of the Murray-von Neumann hyperfinite factor [math]R[/math].


As already mentioned on several occasions, this longstanding question is in need of some new, brave functional analysis input, in relation with the notion of hyperfiniteness, which is probably of quite difficult type, beyond what the current experts can do. \begin{exercises} Congratulations for having read this book, and no exercises here. But, as mentioned above, some good, difficult questions regarding [math]R[/math] are waiting for input from you. \begin{thebibliography}{99} \baselineskip=13.5pt \bibitem{agz}G.W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge Univ. Press (2010). \bibitem{arn}V.I. Arnold, Mathematical methods of classical mechanics, Springer (1974). \bibitem{arv}W. Arveson, An invitation to C[math]^*[/math]-algebras, Springer (1976). \bibitem{aha}M. Asaeda and U. Haagerup, Exotic subfactors of finite depth with Jones indices [math](5+\sqrt{13})/2[/math] and [math](5+\sqrt{17})/2[/math], Comm. Math. 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General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].

References

  1. T. Banica and B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), 277--302.
  2. T. Banica and S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1--14.
  3. T. Banica and D. Goswami, Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010), 343--356.
  4. H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. 149 (1999), 1023--1060.
  5. B. Collins and P. \'Sniady, Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys. 264 (2006), 773--795.
  6. P. Di Francesco, Meander determinants, Comm. Math. Phys. 191 (1998), 543--583.
  7. Z. Liu, S. Morrison and D. Penneys, 1-supertransitive subfactors with index at most [math]6\frac{1}{5}[/math], Comm. Math. Phys. 334 (2015), 889--922.
  8. A. Nica and R. Speicher, Lectures on the combinatorics of free probability, Cambridge Univ. Press (2006).
  9. P. Tarrago and J. Wahl, Free wreath product quantum groups and standard invariants of subfactors, Adv. Math. 331 (2018), 1--57.
  10. H. Wenzl, [math]{\rm C}^*[/math]-tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), 261--282.
  11. V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1--25.