Completely isometric representations of $M_{cb}A(G)$ and $UCB(\hat G)^*$
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- by Matthias Neufang, Zhong-Jin Ruan and Nico Spronk PDF
- Trans. Amer. Math. Soc. 360 (2008), 1133-1161 Request permission
Abstract:
Let $G$ be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra $M_{cb}A(G)$, which is dual to the representation of the measure algebra $M(G)$, on $\mathcal {B}(L_2(G))$. The image algebras of $M(G)$ and $M_{cb}A(G)$ in $\mathcal {CB}^{\sigma } (\mathcal {B}(L_2(G)))$ are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group $G$, there is a natural completely isometric representation of $UCB(\hat G)^*$ on $\mathcal {B}(L_2(G))$, which can be regarded as a duality result of Neufang’s completely isometric representation theorem for $LUC(G)^*$.References
- B. Brainerd and R. E. Edwards, Linear operators which commute with translations. I. Representation theorems, J. Austral. Math. Soc. 6 (1966), 289–327. MR 0206725
- David P. Blecher and Roger R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992), no. 1, 126–144. MR 1157556, DOI 10.1112/jlms/s2-45.1.126
- Marek Bożejko, Positive definite bounded matrices and a characterization of amenable groups, Proc. Amer. Math. Soc. 95 (1985), no. 3, 357–360. MR 806070, DOI 10.1090/S0002-9939-1985-0806070-2
- Marek Bożejko and Gero Fendler, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2, 297–302 (English, with Italian summary). MR 753889
- Philip C. Curtis Jr. and Alessandro Figà-Talamanca, Factorization theorems for Banach algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 169–185. MR 0203500
- Edward G. Effros and Ruy Exel, On multilinear double commutant theorems, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 81–94. MR 996441
- 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
- Edward G. Effros, Jon Kraus, and Zhong-Jin Ruan, On two quantized tensor products, Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991) Res. Notes Math., vol. 5, A K Peters, Wellesley, MA, 1993, pp. 125–145. MR 1259063
- Edward G. Effros and Zhong-Jin Ruan, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988), no. 1, 137–158. MR 950830
- Edward G. Effros and Zhong-Jin Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), no. 2, 579–584. MR 1163332, DOI 10.1090/S0002-9939-1993-1163332-4
- 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, Operator space tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003), no. 1, 131–156. MR 2015023
- Michel Enock and Jean-Marie Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992. With a preface by Alain Connes; With a postface by Adrian Ocneanu. MR 1215933, DOI 10.1007/978-3-662-02813-1
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
- F. Ghahramani, Isometric representation of $M(G)$ on $B(H)$, Glasgow Math. J. 23 (1982), no. 2, 119–122. MR 663137, DOI 10.1017/S0017089500004882
- Fereidoun Ghahramani and Anthony To-Ming Lau, Multipliers and modulus on Banach algebras related to locally compact groups, J. Funct. Anal. 150 (1997), no. 2, 478–497. MR 1479549, DOI 10.1006/jfan.1997.3133
- F. Ghahramani, A. T. Lau, and V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 273–283. MR 1005079, DOI 10.1090/S0002-9947-1990-1005079-2
- J.E. Gilbert, $L^p$-convolution operators of Banach space tensor products I, II, III. Unpublished manuscript.
- Edmond E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371–382. MR 336241, DOI 10.1090/S0002-9947-1974-0336241-0
- U. Haagerup, Decomposition of completely bounded maps on operator algebras. Unpublished manuscript 1980.
- U. Haagerup, Group $C^*$-algebras without the completely bounded approximation property. Unpublished manuscript 1986.
- Uffe Haagerup and Jon Kraus, Approximation properties for group $C^*$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699. MR 1220905, DOI 10.1090/S0002-9947-1994-1220905-3
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Helmut Hofmeier and Gerd Wittstock, A bicommutant theorem for completely bounded module homomorphisms, Math. Ann. 308 (1997), no. 1, 141–154. MR 1446204, DOI 10.1007/s002080050069
- Paul Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), no. 2, 311–313. MR 1180643, DOI 10.4064/cm-63-2-311-313
- Jon Kraus and Zhong-Jin Ruan, Multipliers of Kac algebras, Internat. J. Math. 8 (1997), no. 2, 213–248. MR 1442436, DOI 10.1142/S0129167X9700010X
- Anthony To Ming Lau, Operators which commute with convolutions on subspaces of $L_{\infty }(G)$, Colloq. Math. 39 (1978), no. 2, 351–359. MR 522378, DOI 10.4064/cm-39-2-351-359
- Anthony To Ming Lau, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 251 (1979), 39–59. MR 531968, DOI 10.1090/S0002-9947-1979-0531968-4
- V. Losert, Fourier-Algebra und mittelbare Gruppen. Lectures at Heidelberg University, 1985.
- Bojan Magajna, On completely bounded bimodule maps over $W^*$-algebras, Studia Math. 154 (2003), no. 2, 137–164. MR 1949927, DOI 10.4064/sm154-2-3
- M. Neufang, Abstrakte harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren. Ph.D. thesis at University of Saarland, Saarbrücken, Germany, 2000.
- M. Neufang, Isometric representations of convolution algebras as completely bounded module homomorphisms and a characterization of the measure algebra. Preprint.
- M. Neufang, Bicommutant theorems in the operator space $\mathcal {CB}(\mathcal {B}(H))$, II – Automatic normality and a non-commutative version of the Brainerd–Edwards Theorem. Preprint.
- Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
- G. Pisier, The similarity degree of an operator algebra, Algebra i Analiz 10 (1998), no. 1, 132–186; English transl., St. Petersburg Math. J. 10 (1999), no. 1, 103–146. MR 1618400
- Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
- Zhong-Jin Ruan, Subspaces of $C^*$-algebras, J. Funct. Anal. 76 (1988), no. 1, 217–230. MR 923053, DOI 10.1016/0022-1236(88)90057-2
- Zhong-Jin Ruan, The operator amenability of $A(G)$, Amer. J. Math. 117 (1995), no. 6, 1449–1474. MR 1363075, DOI 10.2307/2375026
- R. R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), no. 1, 156–175. MR 1138841, DOI 10.1016/0022-1236(91)90139-V
- Allan M. Sinclair and Roger R. Smith, Hochschild cohomology of von Neumann algebras, London Mathematical Society Lecture Note Series, vol. 203, Cambridge University Press, Cambridge, 1995. MR 1336825, DOI 10.1017/CBO9780511526190
- N. Spronk, On multipliers of the Fourier algebra of a locally compact group. Ph.D. thesis at University of Waterloo, Ontario, Canada, 2002.
- Nico Spronk, Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. London Math. Soc. (3) 89 (2004), no. 1, 161–192. MR 2063663, DOI 10.1112/S0024611504014650
- W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
- Erling Størmer, Regular abelian Banach algebras of linear maps of operator algebras, J. Functional Analysis 37 (1980), no. 3, 331–373. MR 581427, DOI 10.1016/0022-1236(80)90048-8
- J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911
Additional Information
- Matthias Neufang
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- MR Author ID: 718390
- Email: mneufang@math.carleton.ca
- Zhong-Jin Ruan
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 249360
- Email: ruan@math.uiuc.edu
- Nico Spronk
- Affiliation: Department of Mathematics, University of Walterloo, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 671665
- Email: nspronk@math.uwaterloo.ca
- Received by editor(s): October 26, 2004
- Received by editor(s) in revised form: December 22, 2004
- Published electronically: October 16, 2007
- Additional Notes: The first and third authors were partially supported by NSERC
The second author was partially supported by the National Science Foundation DMS-0140067 and DMS-0500535
The third author was partially supported by an NSERC PDF - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 1133-1161
- MSC (2000): Primary 22D15, 22D20, 43A10, 43A22, 46L07, 46L10, 47L10
- DOI: https://doi.org/10.1090/S0002-9947-07-03940-2
- MathSciNet review: 2357691