Bounded and completely bounded local derivations from certain commutative semisimple Banach algebras
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- by Ebrahim Samei PDF
- Proc. Amer. Math. Soc. 133 (2005), 229-238 Request permission
Abstract:
We show that for a locally compact group $G$, every completely bounded local derivation from the Fourier algebra $A(G)$ into a symmetric operator $A(G)$-module or the operator dual of an essential $A(G)$-bimodule is a derivation. Moreover, for amenable $G$ we show that the result is true for all operator $A(G)$-bimodules. In particular, we derive a new proof to a result of N. Spronk that $A(G)$ is always operator weakly amenable.References
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
- Randall L. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996), no. 1, 76–92. MR 1367625, DOI 10.1006/jfan.1996.0004
- H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
- Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
- Edward G. Effros and Zhong-Jin Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), no. 2, 163–187. MR 1060634, DOI 10.1142/S0129167X90000113
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
- Brian Forrest and Peter Wood, Cohomology and the operator space structure of the Fourier algebra and its second dual, Indiana Univ. Math. J. 50 (2001), no. 3, 1217–1240. MR 1871354, DOI 10.1512/iumj.2001.50.1835
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482, DOI 10.5802/aif.473
- B. E. Johnson, Local derivations on $C^*$-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001), no. 1, 313–325. MR 1783788, DOI 10.1090/S0002-9947-00-02688-X
- Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
- David R. Larson and Ahmed R. Sourour, Local derivations and local automorphisms of ${\scr B}(X)$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 187–194. MR 1077437, DOI 10.1090/pspum/051.2/1077437
- Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Viktor Losert, On tensor products of Fourier algebras, Arch. Math. (Basel) 43 (1984), no. 4, 370–372. MR 802314, DOI 10.1007/BF01196662
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
- Nico Spronk, Operator weak amenability of the Fourier algebra, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3609–3617. MR 1920041, DOI 10.1090/S0002-9939-02-06680-7
- Masamichi Takesaki and Nobuhiko Tatsuuma, Duality and subgroups. II, J. Functional Analysis 11 (1972), 184–190. MR 0384995, DOI 10.1016/0022-1236(72)90087-0
- Jun Tomiyama, Tensor products of commutative Banach algebras, Tohoku Math. J. (2) 12 (1960), 147–154. MR 115108, DOI 10.2748/tmj/1178244494
Additional Information
- Ebrahim Samei
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
- Email: umsameie@cc.umanitoba.ca
- Received by editor(s): June 13, 2003
- Received by editor(s) in revised form: September 30, 2003
- Published electronically: July 26, 2004
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 229-238
- MSC (2000): Primary 46L07, 47B47
- DOI: https://doi.org/10.1090/S0002-9939-04-07555-0
- MathSciNet review: 2085174