Matrix coefficients of unitary representations and associated compactifications

Indiana University Mathematics Journal

Volumn 62, Issue 1, 2013, Pages 99-148

  Spronk, Nico ^a   Stokke, Ross ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b UNIVERSITY OF WINNIPEG   (Canada)

Author keywords

Amenable group; Fourier Stieltjes algebras; Semitopological compactification; ilov boundary

Indexed keywords

EID: 84904480084     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2013.62.4825     Document Type: Article

References (64)

1. R. ARENS and I.M. SINGER, Generalized analytic functions, Trans. Amer.Math. Soc. 81 (1956), no. 2, 379-393. http://dx.doi.org/10.2307/1992923. MR0078657 (17,1226e)
2. G. ARSAC, Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), no. 2, 1-101 (French). MR0444833 (56 #3180)
3. M. E. B. BEKKA, Amenable unitary representations of locally compact groups, Invent. Math. 100 (1990), no. 2, 383-401. http://dx.doi.org/10.1007/ BF01231192. MR1047140 (91g:22007)
4. M. E. B. BEKKA, A. T.M. LAU, and G. SCHLICHTING, On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group, Math. Ann. 294 (1992), no. 3, 513-522. http://dx.doi.org/10.1007/BF01934339. MR1188135 (93k:43006)
5. J. F. BERGLUND, Compact semitopological inverse Clifford semigroups, Semigroup Forum 5 (1972/73), no. 1, 191-215. http://dx.doi.org/10.1007/ BF02572892. MR0325835 (48 #4181)
6. J. F. BERGLUND, H. D. JUNGHENN, and P. MILNES, Analysis on Semigroups: Function Spaces, Compactifications, Representations, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1989. A Wiley-Interscience Publication. MR999922 (91b:43001)
7. R. B. BURCKEL, Weakly Almost Periodic Functions on Semigroups, Gordon and Breach Science Publishers, New York, 1970. MR0263963 (41 #8562)
8. J. DE CANNIERE and U. HAAGERUP, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455-500. http://dx.doi.org/10.2307/2374423. MR784292 (86m:43002)
9. D. I. CARTWRIGHT and J. R. MCMULLEN, A generalized universal complexification for compact groups, J. Reine Angew. Math. 331 (1982), 1-15. http://dx.doi.org/10.1515/crll.1982.331.1. MR647370 (84d:22009)
10. C. CHOU, Minimally weakly almost periodic groups, J. Funct. Anal. 36 (1980), no. 1, 1-17. http://dx.doi.org/10.1016/0022-1236(80)90103-2. MR568972 (81f:43009)
11. -Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), no. 1, 141-157. http://dx.doi.org/10.2307/1999501. MR670924 (84a:43008)
12. M. COWLING, The Fourier-Stieltjes algebra of a semisimple group, Colloq.Math. 41 (1979), no. 1, 89-94. MR550633 (81e:43005)
13. 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. MR1816726 (2002e:46001)
14. J. DIXMIER, C*-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing Co., Amsterdam, 1977. Translated from the French by Francis Jellett. MR0458185 (56 #16388)
15. W. F. EBERLEIN, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer.Math. Soc. 67 (1949), no. 1, 217-240. http://dx.doi.org/10.2307/1990424. MR0036455 (12,112a)
16. E. ELGUN, Idempotents in the Eberlein compactifications of locally compact abelian groups (2011), preprint.
17. P. EYMARD, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236 (French). MR0228628 (37 #4208)
18. H. FARHADI and F. GHAHRAMANI, Involutions on the second duals of group algebras and a multiplier problem, Proc. Edinb. Math. Soc. (2) 50 (2007), no. 1, 153-161. http://dx.doi.org/10.1017/S0013091505000660. MR2294009 (2008b:43004)
19. J. FARAUT, Analysis on Lie Groups: An Introduction, Cambridge Studies in Advanced Mathematics, vol. 110, Cambridge University Press, Cambridge, 2008. http://dx.doi.org/10.1017/CBO9780511755170. MR2426516 (2009f:22009)
20. J.M. G. FELL, Weak containment and induced representations of groups, Canad. J. Math. 14 (1962), 237-268. http://dx.doi.org/10.4153/CJM-1962-016-6. MR0150241 (27 #242)
21. G. B. FOLLAND, A course in Abstract Harmonic Analysis, Studies in AdvancedMathematics, CRC Press, Boca Raton, FL, 1995. MR1397028 (98c:43001)
22. B. E. FORREST, V. RUNDE, and N. SPRONK, Operator amenability of the Fourier algebra in the cb-multiplier norm, Canad. J. Math. 59 (2007), no. 5, 966-980. http://dx.doi.org/10.4153/CJM-2007-041-9. MR2354398 (2008k:43005)
23. Th.W. GAMELIN, Uniform Algebras, Prentice-Hall Inc., Englewood Cliffs, N. J., 1969. MR0410387 (53 #14137)
24. F. GHAHRAMANI, A. T.M. LAU, and V. LOSERT, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 273-283. http://dx.doi.org/10.2307/2001602. MR1005079 (90m:43010)
25. I. GLICKSBERG, Weak compactness and separate continuity, Pacific J. Math. 11 (1961), no. 1, 205-214. http://dx.doi.org/10.2140/pjm.1961.11.205. MR0120523 (22 #11275)
26. C. C. GRAHAM and O. C. MCGEHEE, Essays in Commutative Harmonic Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, Springer-Verlag, New York, 1979. MR550606 (81d:43001)
27. S. A. GRIGORYAN and T. V. TONEV, Shift-invariant Uniform Algebras on Groups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 68, Birkhäuser Verlag, Basel, 2006. MR2239563 (2007m:46074)
28. A. GROTHENDIECK, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), no. 1, 168-186 (French). http://dx.doi.org/10.2307/2372076. MR0047313 (13,857e)
29. U. HAAGERUP, M0A(G) functions which are not coefficients of uniformly bounded representations (1985), unpublished manuscrpit.
30. HARISH-CHANDRA, Plancherel formula for the 2 ?2 real unimodular group, Proc. Nat. Acad. Sci. U. S. A. 38 (1952), no. 4, 337-342. http://dx.doi.org/10. 1073/pnas.38.4.337. MR0047055 (13,820h)
31. E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups, Integration Theory, Group Representations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin, 1979. MR551496 (81k:43001)
32. -Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Die Grundlehren der mathematischenWissenschaften, Band 152, Springer-Verlag, New York, 1970. MR0262773 (41 #7378)
33. R. E. HOWE and C. C. MOORE, Asymptotic properties of unitary representations, J. Funct. Anal. 32 (1979), no. 1, 72-96. http://dx.doi.org/10. 1016/0022-1236(79)90078-8. MR533220 (80g:22017)
34. R. E. HOWE and E. C. TAN, Nonabelian Harmonic Analysis: Applications of SL(2,R), Universitext, Springer-Verlag, New York, 1992. http://dx.doi.org/10. 1007/978-1-4613-9200-2. MR1151617 (93f:22009)
35. A. HULANICKI, Groups whose regular representation weakly contains all unitary representations, Studia Math. 24 (1964), 37-59. MR0191998 (33 #225)
36. M. ILIE and N. SPRONK, The spine of a Fourier-Stieltjes algebra, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 273-301. http://dx.doi.org/10.1112/plms/ pdl002. MR2308229 (2008k:43007)
37. -Corrigendum: The spine of a Fourier-Stieltjes algebra [MR 2308229], Proc. Lond. Math. Soc. (3) 104 (2012), no. 4, 859-863. http://dx.doi.org/10. 1112/plms/pdr057. MR2908785
38. E. KANIUTH, A Course in Commutative Banach Algebras, Graduate Texts in Mathematics, vol. 246, Springer, New York, 2009. http://dx.doi.org/10.1007/978- 0-387-72476-8. MR2458901 (2010d:46064)
39. Y. KATZNELSON, An Introduction to Harmonic Analysis, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. MR2039503 (2005d:43001)
40. I. KHALIL, Sur l'analyse harmonique du groupe affine de la droite, Studia Math. 51 (1974), 139-167 (French). MR0350330 (50 #2823)
41. R. A. KUNZE and E.M. STEIN, Uniformly bounded representations and harmonic analysis of the 2 ?2 real unimodular group, Amer. J. Math. 82 (1960), no. 1, 1-62. http://dx.doi.org/10.2307/2372876. MR0163988 (29 #1287)
42. A. T.M. LAU, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 273-283. http://dx.doi.org/10.1017/S0305004100064197. MR817669 (87i:43001)
43. A. T.M. LAU and R. J. LOY, Weak amenability of Banach algebras on locally compact groups, J. Funct. Anal. 145 (1997), no. 1, 175-204. http://dx.doi.org/10.1006/jfan.1996.3002. MR1442165 (98b:46068)
44. J. LUDWIG, N. SPRONK, and L. TUROWSKA, Beurling-Fourier algebras on compact groups: spectral theory, J. Funct. Anal. 262 (2012), no. 2, 463-499. http://dx.doi.org/10.1016/j.jfa.2011.09.017. MR2854710
45. M. MAYER, Asymptotics of matrix coefficients and closures of Fourier-Stieltjes algebras, J. Funct. Anal. 143 (1997), no. 1, 42-54. http://dx.doi.org/10.1006/jfan.1996.2965. MR1428116 (98d:22004)
46. -Strongly mixing groups, Semigroup Forum 54 (1997), no. 3, 303-316. http://dx.doi.org/10.1007/BF02676613. MR1436849 (98k:43012)
47. K. MCKENNON, The structure space of the trigonometric polynomials on a compact group, J. Reine Angew. Math. 307/308 (1979), 166-172. http://dx.doi.org/10.1515/crll.1979.307-308.166. MR534218 (80j:22003)
48. M.MEGRELISHVILI, Reflexively representable but not Hilbert representable compact flows and semitopological semigroups, Colloq. Math. 110 (2008), no. 2, 383-407. http://dx.doi.org/10.4064/cm110-2-5. MR2353912 (2008k:54062)
49. D. MILICIC, Topological representation of the group Calgebra of SL(2,R), GlasnikMat. Ser. III 6(26) (1971), 231-246 (English, with Serbo-Croatian summary). MR0308795 (46 #7909)
50. D. MUMFORD, The Red Book of Varieties and Schemes, Lecture Notes inMathematics, vol. 1358, Springer-Verlag, Berlin, 1988. MR971985 (89k:14001)
51. M. NEUFANG, A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis, Arch. Math. (Basel) 82 (2004), no. 2, 164-171. http://dx.doi.org/10.1007/s00013-003-0516-7. MR2047670 (2005g:22004)
52. Th.W. PALMER, Banach Algebras and the General Theory of-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR1270014 (95c:46002)
53. G. PISIER, Are unitarizable groups amenable?, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 323-362. http://dx.doi.org/10.1007/3-7643-7447-0 8. MR2195457 (2007e:43001)
54. J. S. PYM, The convolution of linear functionals, Proc. London Math. Soc. (3) 14 (1964), no. 3, 431-444. http://dx.doi.org/10.1112/plms/s3-14.3.431. MR0176342 (31 #617)
55. W. RUDIN, Weak almost periodic functions and Fourier-Stieltjes transforms, Duke Math. J. 26 (1959), no. 2, 215-220. http://dx.doi.org/10.1215/ S0012-7094-59-02622-5. MR0102705 (21 #1492)
56. -Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, vol. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR0152834 (27 #2808)
57. V. RUNDE and N. SPRONK, Operator amenability of Fourier-Stieltjes algebras, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 675-686. http://dx.doi.org/10.1017/S030500410300745X. MR2055055 (2005b:46106)
58. R. STOKKE, Amenable representations and coefficient subspaces of Fourier-Stieltjes algebras, Math. Scand. 98 (2006), no. 2, 182-200. MR2243701 (2007b:22006)
59. -Homomorphisms of convolution algebras, J. Funct. Anal. 261 (2011), no. 12, 3665-3695. http://dx.doi.org/10.1016/j.jfa.2011.08.014. MR2838038 (2012h:43007)
60. A. ÜLGER, Continuity of weakly almost periodic functionals on L1(G), Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 495-497. http://dx.doi.org/10.1093/qmath/37.4.495. MR868624 (88b:43005)
61. W. A. VEECH, Weakly almost periodic functions on semisimple Lie groups, Monatsh. Math. 88 (1979), no. 1, 55-68. http://dx.doi.org/10.1007/BF01305857. MR550072 (81b:22012)
62. G. XU, Herz-Schur multipliers and weakly almost periodic functions on locally compact groups, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2525-2536. http://dx.doi.org/10.1090/S0002-9947-97-01733-9. MR1373647 (97h:43003)
63. M. E. WALTER, W * *-algebras and nonabelian harmonic analysis, J. Functional Analysis 11 (1972), no. 1, 17-38. http://dx.doi.org/10. 1016/0022-1236(72)90077-8.MR0352879 (50 #5365)
64. -On the structure of the Fourier-Stieltjes algebra, Pacific J. Math. 58 (1975), no. 1, 267-281. http://dx.doi.org/10.2140/pjm.1975.58.267. MR0425008 (54 #12966) Canada Canada