A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal
HTML articles powered by AMS MathViewer
- by Volker Runde PDF
- Trans. Amer. Math. Soc. 358 (2006), 391-402 Request permission
Erratum: Trans. Amer. Math. Soc. 367 (2015), 751-754.
Abstract:
Let $G$ be a locally compact group, and let $\mathcal {WAP}(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[\operatorname {SIN}]$-group, but not compact, then the dual Banach algebra $\mathcal {WAP}(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[\operatorname {SIN}]$-group, $\mathcal {WAP}(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups $G$ such that $\mathcal {WAP}(G)^\ast$ does have a normal, virtual diagonal.References
- John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
- John W. Bunce and William L. Paschke, Quasi-expectations and amenable von Neumann algebras, Proc. Amer. Math. Soc. 71 (1978), no. 2, 232–236. MR 482252, DOI 10.1090/S0002-9939-1978-0482252-3
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Ching Chou, Minimally weakly almost periodic groups, J. Functional Analysis 36 (1980), no. 1, 1–17. MR 568972, DOI 10.1016/0022-1236(80)90103-2
- Ching Chou, Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), no. 1, 141–157. MR 670924, DOI 10.1090/S0002-9947-1982-0670924-2
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- A. Connes, On the cohomology of operator algebras, J. Functional Analysis 28 (1978), no. 2, 248–253. MR 0493383, DOI 10.1016/0022-1236(78)90088-5
- Gustavo Corach and José E. Galé, Averaging with virtual diagonals and geometry of representations, Banach algebras ’97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 87–100. MR 1656600
- 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. MR 1911870, DOI 10.1112/S0024610702003381
- Edward G. Effros, Amenability and virtual diagonals for von Neumann algebras, J. Funct. Anal. 78 (1988), no. 1, 137–153. MR 937636, DOI 10.1016/0022-1236(88)90136-X
- Edward G. Effros and E. Christopher Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1–34. MR 448092, DOI 10.1016/0001-8708(77)90085-8
- Edward G. Effros and Akitaka Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36 (1987), no. 2, 257–276. MR 891774, DOI 10.1512/iumj.1987.36.36015
- S. Ferri and D. Strauss, A note on the $\scr {WAP}$-compactification and the $\scr {LUC}$-compactification of a topological group, Semigroup Forum 69 (2004), no. 1, 87–101. MR 2063981, DOI 10.1007/s00233-003-0026-8
- A. Ya. Khelemskiĭ, Homological essence of amenability in the sense of A. Connes: the injectivity of the predual bimodule, Mat. Sb. 180 (1989), no. 12, 1680–1690, 1728 (Russian); English transl., Math. USSR-Sb. 68 (1991), no. 2, 555–566. MR 1038222, DOI 10.1070/SM1991v068n02ABEH001374
- B. E. Johnson, Separate continuity and measurability, Proc. Amer. Math. Soc. 20 (1969), 420–422. MR 236345, DOI 10.1090/S0002-9939-1969-0236345-0
- Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
- B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685–698. MR 317050, DOI 10.2307/2373751
- B. E. Johnson, R. V. Kadison, and J. R. Ringrose, Cohomology of operator algebras. III. Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972), 73–96. MR 318908
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- Volker Runde, Amenability for dual Banach algebras, Studia Math. 148 (2001), no. 1, 47–66. MR 1881439, DOI 10.4064/sm148-1-5
- Volker Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. MR 1874893, DOI 10.1007/b82937
- Volker Runde, Connes-amenability and normal, virtual diagonals for measure algebras. I, J. London Math. Soc. (2) 67 (2003), no. 3, 643–656. MR 1967697, DOI 10.1112/S0024610703004125
- Volker Runde, Connes-amenability and normal, virtual diagonals for measure algebras. II, Bull. Austral. Math. Soc. 68 (2003), no. 2, 325–328. MR 2016307, DOI 10.1017/S0004972700037709
- Volker Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand. 95 (2004), no. 1, 124–144. MR 2091485, DOI 10.7146/math.scand.a-14452
- Simon Wassermann, On tensor products of certain group $C^{\ast }$-algebras, J. Functional Analysis 23 (1976), no. 3, 239–254. MR 0425628, DOI 10.1016/0022-1236(76)90050-1
- Simon Wassermann, Injective $W^*$-algebras, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 1, 39–47. MR 448108, DOI 10.1017/S0305004100053664
Additional Information
- Volker Runde
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- Email: vrunde@ualberta.ca
- Received by editor(s): October 26, 2003
- Received by editor(s) in revised form: June 1, 2004
- Published electronically: July 26, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 391-402
- MSC (2000): Primary 46H20; Secondary 22A15, 22A20, 43A07, 43A10, 43A60, 46H25, 46M18, 46M20
- DOI: https://doi.org/10.1090/S0002-9947-05-03827-4
- MathSciNet review: 2171239