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The commutant of $L\left(H\right)$ in its ultrapower may or may not be trivial. (English) Zbl 1200.46049
Summary: E. Kirchberg [Abel Symposia 1, 175–231 (2006; Zbl 1118.46054)] asked whether the commutant of $L\left(H\right)$ in its (norm) ultrapower is trivial. Assuming the continuum hypothesis, we prove that the answer depends on the choice of the ultrafilter.
MSC:
 46L05 General theory of ${C}^{*}$-algebras 03E50 Continuum hypothesis; Martin’s axiom (logic)
Keywords:
commutant; ultrapower
References:
 [1] Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Magidor, M., Kanamori, A. (eds.) Handbook of Set Theory, Available at http://www.math.lsa.umich.edu/$\sim$ablass/set.html [2] Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In: Chatzidakis, Z. et al. (eds.) Model Theory with Applications to Algebra and Analysis, vol. II. Lecture Notes series of the London Math. Society, no. 350. Cambridge University Press, pp. 315–427 (2008) [3] Connes A.: Outer conjugacy classes of automorphisms of factors. Ann. Sci. Éc. Norm. Sup. Sér. 4(8), 383–420 (1975) [4] Dixmier J.: Von Neumann Algebras. North-Holland, Amsterdam (1981) [5] Farah, I.: All automorphisms of the Calkin algebra are inner. preprint (arXiv: 0705.3085v7 [math.OA]) [6] Farah I.: Semiselective coideals. Mathematika 45, 79–103 (1998) · Zbl 0903.03029 · doi:10.1112/S0025579300014054 [7] Farah I.: The relative commutant of separable C*-algebras of real rank zero. J. Funct. Anal. 256, 3841–3846 (2009) · Zbl 1177.46042 · doi:10.1016/j.jfa.2008.10.007 [8] Farah, I., Hart, B., Sherman, D.: Model theory of operator algebras I: stability. preprint (arXiv: 0908. 2790v1 [math.OA]) [9] Farah, I., Hart, B., Sherman, D.: Model theory of operator algebras II: model theory (2009, in preparation) [10] Farah, I., Steprāns, J.: Flat ultrafilters (2009, in preparation) [11] Ge, L., Hadwin, D.: Ultraproducts of C*-algebras. Recent advances in operator theory and related topics (Szeged, 1999), Oper. Theory Adv. Appl. vol. 127. Birkhäuser, Basel, pp. 305–326 (2001) [12] Izumi M.: Finite group actions on C*-algebras with the Rohlin property. I. Duke Math. J. 122, 233–280 (2004) · Zbl 1067.46058 · doi:10.1215/S0012-7094-04-12221-3 [13] Jones, V.F.R.: Actions of finite groups on the hyperfinite type II1 factor. Mem. Am. Math. Soc. 28, 237 (1980) [14] Kirchberg, E.: The classification of purely infinite C*-algebras using Kasparov’s theory. preliminary preprint (3rd draft) [15] Kirchberg, E.: Central sequences in C*-algebras and strongly purely infinite algebras. Operator algebras: the Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, pp. 175–231 (2006) [16] Kirchberg E., Phillips N.C.: Embedding of exact C*-algebras in the Cuntz algebra ${𝒪}_{2}$ . J. Reine Angew. Math. 525, 17–53 (2000) · Zbl 0973.46048 · doi:10.1515/crll.2000.065 [17] Kunen K.: Some points in $\beta$ N. Math. Proc. Cambridge Philos. Soc. 80(3), 385–398 (1976) · Zbl 0345.02047 · doi:10.1017/S0305004100053032 [18] Kunen K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980) [19] Mathias A.R.D.: Happy families. Ann. Math. Logic 12, 59–111 (1977) · Zbl 0369.02041 · doi:10.1016/0003-4843(77)90006-7 [20] McDuff D.: Central sequences and the hyperfinite factor. Proc. Lond. Math. Soc. 21, 443–461 (1970) · Zbl 0204.14902 · doi:10.1112/plms/s3-21.3.443 [21] Ocneanu, A.: Actions of Discrete Amenable Groups on von Neumann Algebras. Springer-Verlag Lecture Notes in Math. no. 1138. Springer, Berlin (1985) [22] Phillips N.C.: A classification theorem for nuclear purely infinite simple C*-algebras. Documenta Math. 5, 49–114 (2000) (electronic) [23] Phillips N.C., Weaver N.C.: The Calkin algebra has outer automorphisms. Duke Math. J. 139, 185–202 (2007) · Zbl 1220.46040 · doi:10.1215/S0012-7094-07-13915-2 [24] Shelah, S.: Proper and Improper Forcing, 2nd edn. Perspectives in Mathematical Logic, Springer, Berlin (1998) [25] Sherman, D.: Divisible operators in von Neumann algebras. Illinois J. Math. (in press) [26] Takesaki M.: Theory of Operator Algebras III. Springer, Berlin (2003)