Abstract
In this paper, it is proved that the predual bimodule of the measure algebra of an infinite discrete group is not injective despite the fact that the measure algebra of an amenable group is amenable in the sense of Connes. Thus the well-known result of Khelemskii (claiming that, for a von Neumann algebra, Connes-amenability is equivalent to the condition that the predual bimodule is injective) cannot be extended to measure algebras. Moreover, for a discrete amenable group, we give a simple formula for a normal virtual diagonal of the measure algebra. It is shown that a certain canonical bimodule over the measure algebra is not normal.
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REFERENCES
B. E. Johnson, “Cohomology in Banach algebras,” Mem. Amer. Math. Soc., 127 (1972).
V. Runde, Lectures on Amenability. Lecture Notes in Math., vol. 1774, Springer-Verlag, Berlin, 2002.
A. Connes, “On the Cohomology of Operator Algebras,” J. Funct. Anal., 28 (1978), 248–253.
U. Haagerup, “All nuclear algebras are amenable,” Invent. Math., 74 (1985), 305–316.
J. W. Bunce and W. L. Paschke, “Quasi-expectations and amenable von Neumann algebras,” Proc. Amer. Math. Soc., 71 (1978), 232–236.
V. Runde, “Amenability for dual Banach algebras,” Studia Math., 148 (2001), 47–66.
V. Runde, “Connes-amenability and normal, virtual diagonals for measure algebras, I” J. London Math. Soc. (to appear).
G. Corach and J. E. Galé, “Averaging with virtual diagonals and geometry of representations,” in: Banach Algebras '97 (E. Albrecht and M. Mathieu, editors), Walter de Gruyter, 1998, pp. 87–100.
H. G. Dales, F. Ghahramani, and A. Ya. Helemskii, “The amenability of measure algebras,” J. London Math. Soc. (2) 66 (2002), no. 1, 213–226.
S. Wassermann, “On tensor products of certain group C*-algebras,” J. Funct. Anal., 23 (1976), 239–254.
B. E. Johnson and R. V. Kadison, and J. Ringrose, “Cohomology of operator algebras. III. Reduction to normal cohomology,” Bull. Soc. Math. France, 100 (1972), 73–96.
R. V. Kadison and J. Ringrose, “Cohomology of operator algebras. I. Type I von Neumann algebras,” Acta Math., 126 (1971), 227–243.
R. V. Kadison and J. Ringrose, “Cohomology of operator algebras. II,” Ark. Math., 9 (1971), 55–63.
G. A. Elliott, “On approximately finite-dimensional von Neumann algebras. II,” Canad. Math. Bull., 21 (1978), 415–418.
A. Connes, “Classification of injective factors. Cases II 1, II ∞, III λ, λ ≠ 1,” Ann. of Math., 104 (1976), no. 1, 73–115.
A. Ya. Khelemskii, “Homological essence of amenability in the sense of A. Connes: the injectivity of the predual bimodule,” Mat. Sb. [Math. USSR-Sb.], 180 (1989), no. 12, 1680–1690.
A. Ya. Khelemskii, Homology in Banach and Topological Algebras [in Russian], Moskov. Gos. Univ., Moscow, 1986; English transl. in: A. Ya. Helemskii, The Homology of Banach and Topological Algebras (Mathematics and its Applications (Soviet Series), 41), Kluwer Academic Publishers Group, Dordrecht, 1989.
R. S. Phillips, “On linear transformations,” Trans. Amer. Math. Soc., 48 (1940), 516–541.
E. G. Effros, “Amenability and virtual diagonals for von Neumann algebras,” J. Funct. Anal., 78 (1988), 137–153.
V. Runde, Connes-Amenability and Normal, Virtual Diagonals for Measure Algebras. II, Preprint, 2001.
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Tabaldyev, S.B. Noninjectivity of the Predual Bimodule of the Measure Algebra for Infinite Discrete Groups. Mathematical Notes 73, 690–696 (2003). https://doi.org/10.1023/A:1024016906227
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DOI: https://doi.org/10.1023/A:1024016906227