Rocky Mountain Journal of Mathematics

Semi-cosimplicial objects and spreadability

D. Gwion Evans, Rolf Gohm, and Claus Köstler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

To a semi-cosimplicial object (SCO) in a category, we associate a system of partial shifts on the inductive limit. We show how to produce an SCO from an action of the infinite braid monoid $\mathbb{B} ^+_\infty $ and provide examples. In categories of (noncommutative) probability spaces, SCOs correspond to spreadable sequences of random variables; hence, SCOs can be considered as the algebraic structure underlying spreadability.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 6 (2017), 1839-1873.

Dates
First available in Project Euclid: 21 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1511254955

Digital Object Identifier
doi:10.1216/RMJ-2017-47-6-1839

Mathematical Reviews number (MathSciNet)
MR3725247

Zentralblatt MATH identifier
06816573

Subjects
Primary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 20F36: Braid groups; Artin groups 46L53: Noncommutative probability and statistics

Keywords
Semi-cosimplicial object coface operator partial shift braid monoid cohomology noncommutative probability space spreadability subfactor

Citation

Evans, D. Gwion; Gohm, Rolf; Köstler, Claus. Semi-cosimplicial objects and spreadability. Rocky Mountain J. Math. 47 (2017), no. 6, 1839--1873. doi:10.1216/RMJ-2017-47-6-1839. https://projecteuclid.org/euclid.rmjm/1511254955


Export citation

References

  • E. Artin, Theorie der Zöpfe, Abh. Math. Sem. 4, Hamburg University (1925), 47–72.
  • D. Blanc and D. Sen, Mapping spaces and $R$-completion,.
  • S. Curran, A characterization of freeness by invariance under quantum spreading, J. reine angew. Math. 659 (2011), 43–65.
  • K.J. Dykema, C. Köstler and J.D. Williams, Quantum symmetric states on free product $C^*$-algebras, Trans. Amer. Math. Soc. 369 (2017), 645–679.
  • T. Gateva-Ivanova and P. Cameron, Multipermutation solutions of the Yang-Baxter equation, Comm. Math. Phys. 309 (2012), 583–621.
  • R. Gohm, Noncommutative stationary processes, Lect. Notes Math. 1839, Sprinver Verlag, Berlin, 2004.
  • R. Gohm and C. K östler, Noncommutative independence from the braid group $\Bset_\infty$, Comm. Math. Phys. 289 (2009), 435–482.
  • ––––, Noncommutative independence in the infinite braid and symmetric group, Noncommutative harmonic analysis with applications to probability III, Banach Center Publ. 96, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 2012.
  • F. Goodman, P. de la Harpe and V.F.R. Jones, Coxeter graphs and towers of algebras, Springer, Berlin, 1989.
  • O. Kallenberg, Spreading and predictable sampling in exchangeable sequences and processes, Ann. Probab. 16 (1988), 508–534.
  • ––––, Probabilistic symmetries and invariance principles, in Probability and its applications, Springer, New York, 2005.
  • C. Kassel and V. Turaev, Braid groups, Grad. Texts Math. 247, Springer, New York, 2008.
  • C. Köstler, A noncommutative extended de Finetti theorem, J. Funct. Anal. 258 (2010), 1073–1120.
  • S. Mac Lane, Categories for the working mathematician, Springer, New York, 1998.
  • P.A. Mitchener, A primer on some methods in homotopy theory, http://www.mitchener.staff.shef.ac.uk/writing.html, 2003.
  • A. Nica and R. Speicher, Lectures on the combinatorics of free probability, Lect. Notes Math. 335, Cambridge University Press, Cambridge, 2006.
  • K. Petersen, Ergodic theory, Cambridge University Press, Cambridge, 1983.
  • V.A. Smirnov, Simplicial and operad methods in algebraic topology, Transl. Math. Mono. 198, American Mathematical Society, Providence, 2001.
  • C.A. Weibel, An introduction to homological algebra, Cambr. Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994.
  • J. Wu, Simplicial objects and homotopy groups, Braids: Introductory lectures on braids, configurations and their applications, Lect. Note 19, World Scientific, Singapore, 2010.