Skip to main content Accessibility help

Sofic entropy of Gaussian actions

  • BEN HAYES (a1)


Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc. 23 (2010) 217–245], Kerr and Li [Invent. Math. 186 (2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and $C^{\ast }$ -algebras which replace the Fourier analysis used in the abelian case.



Hide All
[1] Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217245.
[2] Bowen, L.. Entropy for expansive algebraic actions of residually finite groups. Ergod. Th. & Dynam. Sys. 31(3) (2011), 703718.
[3] Bowen, L.. Weak isomorphisms between Bernoulli shifts. Israel J. Math. 183 (2011).
[4] Bowen, L.. Every countably infinite group is almost Ornstein. Dynamical Systems and Group Actions. Vol. 567 (Contemporary Mathematics) . American Mathematical Society, Providence, RI, 2012, pp. 6778.
[5] Bowen, L. and Li, H.. Harmonic models and spanning forests of residually finite groups. J. Funct. Anal. 263(7) (2012), 17691808.
[6] Brown, N. and Ozawa, N.. C -Algebras and Finite-Dimensional Approximations. American Mathematical Society, Providence, RI, 1994.
[7] Ciobanu, L., Holt, D. and Rees, S.. Sofic groups: graph products and graphs of groups. Pacific J. Math. 271(1) (2014), 5364.
[8] Conway, J.. A Course in Operator Theory (Graduate Studies in Mathematics) . American Mathematical Society, Providence, RI, 1991.
[9] Dooley, A. and Golodets, V.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196(1) (2002), 118.
[10] Dykema, K., Kerr, D. and Pichot, M.. Orbit equivalence and sofic approximation. Preprint, 2011,arXiv:1102.2556.
[11] Dykema, K., Kerr, D. and Pichot, M.. Sofic dimension for discrete measurable groupoids. Trans. Amer. Math. Soc. 366(2) (2013), 707748.
[12] Elek, G. and Szabo, E.. On sofic groups. J. Group Theory 9(2) (2006), 161171.
[13] Folland, G.. Real Analysis: Modern Techinques and Their Applications, 2nd edn. Wiley, Hoboken, NJ, 1999.
[14] Friedman, J.. A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195 (2008), 1100.
[15] Hayes, B.. Fuglede-Kadison determinants and sofic entropy. Preprint, 2014, arXiv:1402.1135.
[16] Hayes, B.. An l p -version of von Neumann dimension for Banach space representations of sofic groups. J. Funct. Anal. 266(2) (2014), 9891040.
[17] Hayes, B.. Polish models and sofic entropy. J. Inst. Math. Jussieu to appear.
[18] Kerr, D.. Bernoulli actions of sofic groups have completely positive entropy. Israel J. Math. 202(1) (2014), 461474.
[19] Kerr, D. and Li, H.. Topological entropy and the variational principle for actions of sofic groups. Invent. Math. 186 (2011), 501558.
[20] Kieffer, J.. A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space. Ann. Probab. 3(6) (1975), 10311037.
[21] Lemańczyk, M.. Entropy of Gaussian actions for countable abelian groups. Fund. Math. 157 (1998), 277286.
[22] Meyerovitch, T.. Positive sofic entropy implies finite stabilizer. Preprint, 2015, arXiv:1504.08137.
[23] Nica, A.. Asymptotically free families of random unitaries in symmetric groups. Pacific J. Math. 157 (1993), 295310.
[24] Ornstein, D. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1141.
[25] Paunescu, L.. On sofic actions and equivalence relations. J. Funct. Anal. 261(9) (2011), 24612485.
[26] Paunescu, L.. Convex structures revisited. Ergod. Th. & Dynam. Sys. to appear doi:10.1017/etds.2014.129.
[27] Peterson, J. and Sinclair, T.. On cocycle superrigidity for Gaussian actions. Ergod. Th. & Dynam. Sys. 32(1) (2012), 249272.
[28] Popa, S.. Independence properties in subagebras of ultraproduct II1 factors. J. Funct. Anal. 266(9) (2014), 58185846.
[29] Seward, B.. Every action of a non-amenable group is the factor of a small action. J. Mod. Dyn. 8(2) (2014), 251270.
[30] Takesaki, M.. The Theory of Operator Algebras I. Springer, Berlin, 2002.
[31] Takesaki, M.. Theory of Operator Algebras II (Encyclopaedia of Mathematical Sciences, 125) . Springer, New York, 2003.

Related content

Powered by UNSILO

Sofic entropy of Gaussian actions

  • BEN HAYES (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.