Non-commutative entropy computations for continuous fields and cross-products  [ Calculs d’entropie non commutative pour des champs continus ou des produits croisés ]
Annales Mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 1-16.

Nous présentons deux situations de géométrie non commutative dans lesquelles des calculs d’entropie dynamique non commutative sont possibles. La première s’interesse aux automorphismes des produits croisés par un groupe exact qui commutent avec l’action du groupe généralisant un résultat connu pour les groupes moyennables. La seconde concerne les champs continues de C * -algèbres et les automorphismes fibrés. Dans chacun des deux cas, il s’agit d’utiliser des factorisations explicites par des algèbres de matrices.

We present here two non-commutative situations where dynamical entropy estimates are possible. The first result is concerned with automorphisms of cross-products by an exact group that commute with the group action and generalizes the result known for amenable groups. The second is about continuous fields of C * -algebras and C(X)-automorphisms. Each result relies on explicit factorization via matrices.

DOI : https://doi.org/10.5802/ambp.223
Classification : 46L55,  46L53
Mots clés : entropie dynamique non commutative, produits croisés, champs continus
@article{AMBP_2007__14_1_1_0,
     author = {Germain, Emmanuel},
     title = {Non-commutative entropy computations for continuous fields and cross-products},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {1--16},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {1},
     year = {2007},
     doi = {10.5802/ambp.223},
     mrnumber = {2298720},
     zbl = {1132.46041},
     language = {en},
     url = {www.numdam.org/item/AMBP_2007__14_1_1_0/}
}
Germain, Emmanuel. Non-commutative entropy computations for continuous fields and cross-products. Annales Mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 1-16. doi : 10.5802/ambp.223. http://www.numdam.org/item/AMBP_2007__14_1_1_0/

[1] Anantharaman-Delaroche, C.; Renault, J. Amenable Groupoids. With a foreword by Georges Skandalis and Appendix B by E. Germain. Volume 36, L’Enseignement Mathématique, Genève, 2000 | MR 1799683 | Zbl 0960.43003

[2] Anantharaman-Delaroche, Claire Amenability and exactness for dynamical systems and their C * -algebras, Trans. Amer. Math. Soc., Volume 354 (2002), pp. 4153-4178 | Article | MR 1926869 | Zbl 1035.46039

[3] Blanchard, Etienne Déformations de C * -algèbres de Hopf, Bull. Soc. Math. France, Volume 124 (1996), pp. 141-215 | Numdam | MR 1395009 | Zbl 0851.46040

[4] Blanchard, Etienne Subtriviality of continuous fields of nuclear C * -algebras, J. Reine Angew. Math., Volume 498 (1997), pp. 133-149 | Article | MR 1461207 | Zbl 0885.46047

[5] Brown, N. P.; Dykema, K.; Shlyakhtenko, D. Topological entropy of free product automorphisms, Acta Math., Volume 189 (2002), pp. 1-35 | Article | MR 1946916 | Zbl 1021.46050

[6] Brown, Nathanial P. Topological entropy in exact C * -algebras, Math. Ann., Volume 314 (1999), pp. 347-367 | Article | MR 1697449 | Zbl 0952.46039

[7] Brown, Nathaniel P.; Germain, Emmanuel Dual entropy in discrete groups with amenable actions, Ergodic Theory Dynam. Systems, Volume 22 (2002), pp. 711-728 | Article | MR 1908551 | Zbl 1013.37010

[8] Dykema, Kenneth J. Topological entropy of some automorphisms of reduced amalgamated free product C * -algebras, Ergodic Theory Dynam. Systems, Volume 21 (2001), pp. 1683-1693 | Article | MR 1869065 | Zbl 1020.46016

[9] Haagerup, Uffe; Rordam, Mikael Perturbations of the rotation C * -algebras and of the Heisenberg commutation relation, Duke Math. J., Volume 77 (1995), pp. 627-656 | Article | MR 1324637 | Zbl 0830.46058

[10] Kirchberg, Eberhard; Phillips, N. Christopher Embedding of continuous fields of C * -algebras in the Cuntz algebra 𝒪 2 , J. Reine Angew. Math., Volume 525 (2000), pp. 55-94 | Article | MR 1780427 | Zbl 0973.46047

[11] Kirchberg, Eberhard; Wassermann, Simon Exact groups and continuous bundles of C * -algebras, Math. Ann., Volume 315 (1999), pp. 169-203 | Article | MR 1721796 | Zbl 0946.46054

[12] Ozawa, Narutaka Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Ser. I Math., Volume 330 (2000), pp. 691-695 | Article | MR 1763912 | Zbl 0953.43001

[13] Pop, C.; Smith, R. R. Crossed products and entropy of automorphisms, J. Funct. Anal, Volume 206 (2004), pp. 210-232 | Article | MR 2024352 | Zbl 1043.46046

[14] Sinclair, R. R. A. M.and Smith The completely bounded approximation property for discrete crossed products, Indiana Univ. Math. J., Volume 46 (1997), pp. 1311-1322 | Article | MR 1631596 | Zbl 0907.46051

[15] Voiculescu, Dan Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys., Volume 170 (1995), pp. 249-281 | Article | MR 1334396 | Zbl 0824.46079