Representations of Topological Groups in Locally Convex Spaces: Continuity Properties and Weak Almost Periodicity

Russian Journal of Mathematical Physics

Volumn 11, Issue 1, 2004, Pages 81-108

  Shtern, A I ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1842579214     PISSN: 10619208     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (85)

1. Antonyan, S. and Sanchis, M., Extension of Locally Pseudocompact Group Actions, Ann. Mat. Pura Appl. (4), 2002, vol. 181, pp. 239-246.
2. Arendt, W. and Schweiker, S., Discrete Spectrum and Almost Periodicity, Taiwanese J. Math., 1999, vol. 3, no. 4, pp. 475-490.
3. Arhangel'skii, A.V., On a Theorem of W. W. Comfort and K. A. Ross, Comment. Math. Univ. Carolin., 1999, vol. 40, no. 1, pp. 133-151.
4. Arhangel'skii, A. V., Moscow Spaces, Pestov-Tkacenko Problem, and C-embeddings, Comment. Math. Univ. Carolin., 2000, vol. 41, no. 3, pp. 585-595.
5. Banach, S., Théorie des Opérations Linéaires, Subwencji Funduszu Narodowej, Warszawa, 1932.
6. Banaszczyk, W., Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math., 1466, Springer-Verlag, Berlin, 1991.
7. Basit, B. and Emam, M., Differences of Functions in Locally Convex Spaces and Applications to Almost Periodic and Almost Automorphic Functions, Ann. Polon. Math., 1983, vol. 41, no. 3, pp. 193-201.
8. Basit, B., Some Problems Concerning Different Types of Vector Valued Almost Periodic Functions, Dissertationes Math. (Rozprawy Mat.), 1995, vol. 338.
9. Basit, B., Harmonic Analysis and Asymptotic Behavior of Solutions to the Abstract Cauchy Problem, Semigroup Forum, 1997, vol. 54, no. 1, pp. 58-74.
10. Basit, B. and Gunzler, H., Asymptotic Behavior of Solutions of Systems of Neutral and Convolution Equations, J. Differential Equations, 1998, vol. 149, no. 1, pp. 115-142.
11. Basit, B. and Pryde, A. J., Ergodicity and Differences of Functions on Semigroups, J. Austral. Math. Soc. Ser. A, 1998, vol. 64, no. 2, pp. 253-265.
12. Beggs, E. J., Pointwise Bounded Asymptotic Morphisms and Thomsen's Non-Stable k-Theory (to appear).
13. Berglund, J. F., Junghenn, H. D., and Milnes, P., Analysis on Semigroups, 1989, Canadian Math. Soc., A Wiley-Interscience Publication, New York.
14. Brown, L. G., Topologically Complete Groups, Proc. Amer. Math. Soc., 1972, vol. 35, no. 2, pp. 593-600.
15. Brown, L. G., Continuity of Actions of Groups and Semigroups on Banach Spaces, J. London Math. Soc. (2), 2000, vol. 62, no. 1, pp. 107-116.
16. Burckel, R. B., Weakly Almost Periodic Functions on Semigroups, 1970, Gordon and Breach Science Publishers, New York-London-Paris.
17. Chen, Y.-Z. and Lau, K.-S., Harmonic Analysis on Functions with Bounded Means, Commutative harmonic analysis (Canton, NY, 1987), Contemp. Math., 91, Amer. Math. Soc., Providence, RI, 1989, pp. 165-175.
18. Chou C., Weakly Almost Periodic Functions and Almost Convergent Functions on a Group, Trans. Amer. Math. Soc., 1975, vol. 206, pp. 175-200.
19. Chou, C., Weakly Almost Periodic Functions and Fourier-Stieltjes Algebras of Locally Compact Groups, Trans. Amer. Math. Soc., 1982, vol. 274, no. 1, pp. 141-157.
20. Comfort, W. W. and Ross, K. A., Pseudocompactness and Uniform Continuity in Topological Groups, Pacific J. Math., 1966, vol. 16, no. 3, pp. 483-496.
21. Comfort, W. W., Soundararajan, T. and Trigos-Arrieta, F. J., Determining a Weakly Locally Compact Group Topology by its System of Closed Subgroups, Papers in General Topology and Applications, Annals of the New York Academy of Sciences 728 (New York Academy of Sciences, New York, 1994), S. Andina, G. Itzkowitz, T. Y. Kong, R. Kopperman, P. R. Misra, and A. Todd, eds., pp. 25-33.
22. Comfort, W. W. and Trigos-Arrieta, F. J., Locally pseudocompact topological groups, Topology Appl., 1995, vol. 62, no. 3, pp. 263-280.
23. ∞(G) with unique topological left invariant mean, Czechoslovak Math. J., 1984, vol. 34 (109), no. 2, pp. 178-182.
24. Crombez, G., Generalizations of Some Theorems of Loomis on Almost Periodic Functions, Publ. Inst. Math. (Beograd) (N.S.), 1983, vol. 34 (48), pp. 27-29.
25. Cuntz, J., Bivariante K-Theorie für lokalkonvexe Algebren und der bivariante Chern-Connes-Charakter, Doc. Math., 1997, vol. 2, pp. 139-182.
26. Cuntz, J., Bivariant K-Theory and the Weil Algebra (to appear).
27. Davis, W. J., Figiel, T., Johnson, W. B., and Pelczynski, A., Factoring Weakly Compact Operators, J. Funct. Anal., 1974, vol. 17, pp. 311-327.
28. Day, M., Normed Linear Spaces, Third ed., Springer-Verlag, New York-Heidelberg, 1973.
29. Dikranjan, D. and Shakhmatov, D., Algebraic Structure of Pseudocompact Groups, Mem. Amer. Math. Soc. No. 633, American Mathematical Society, Providence, R.I., 1998.
30. Eberlein, W. F., Abstract Ergodic Theorems and Weak Almost Periodic Functions, Trans. Amer. Math. Soc., 1949, vol. 67, pp. 217-240.
31. Eberlein, W. F., The Point Spectrum of Weakly Almost Periodic Functions, Michigan Math. J., 1955-56, vol. 3, pp. 137-139.
32. Edwards, R. E., Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York-Chicago-San Francisco-Toronto-London, 1965.
33. Egghe, L., On the Radon-Nikodym Property, and Related Topics in Locally Convex Spaces, Lecture Notes in Phys., 1978, vol. 77, pp. 77-90.
34. Ellis, R., Locally Compact Transformation Groups, Duke Math. J., 1957, vol. 24, pp. 119-125.
35. Engelking, R., General Topology, Second ed., Heldermann Verlag, Berlin, 1989.
36. Gaal, S.A., Linear Analysis and Representation Theory, Springer-Verlag, New York, 1973.
37. Greenleaf, F. P., Invariant Means on Topological Groups and Their Applications, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969.
38. Grothendieck, A., Critères de compacité dans les espaces fonctioneles généraux, Amer. J. Math., 1952, vol. 74, pp. 168-186.
39. Helmer, D., Joint Continuity of Affine Semigroup Actions, Semigroup Forum, 1980, vol. 21, pp. 153-165.
40. Helmer, D., Continuity of Semigroup Actions, Semigroup Forum, 1981, vol. 23, pp. 153-188.
41. Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, vol. I, Springer-Verlag, Berlin-New York, 1979.
42. Jacobs, K., Ergodentheorie und Fastperiodische Funktionen auf Halbgruppen, Math. Z., 1956, vol. 64, pp. 298-338.
43. Jacobs, K., Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banach-Räumen, Math. Z., 1957, vol. 67, pp. 83-92.
44. Jayne, J. E., Namioka, I., and Rogers, C. A., Topological Properties of Banach Spaces, Proc. London Math. Soc. (3), 1993, vol. 66, no. 3, pp. 651-672.
45. Jayne, J. E. and Rogers, C. A., Borel Selectors for upper Semicontinuous Set-Valued Maps, Acta Math., 1985, vol. 155, pp. 41-79.
46. Johnson, B. E., Cohomology of Banach Algebras, Amer. Math. Soc., Providence, RI, 1972 (Mem. Amer. Math. Soc. V. 127)
47. Junghenn, H. D., Direct Sums of Spaces of Functions on Semigroups, Semigroup Forum, 1994, vol. 49, no. 1, pp. 115-123.
48. Kadets, M. I. and Lyubich, Yu. I., On the Relations Between Different Forms of Almost Periodicity of Representations of Groups, Teor. Funktsii Funktsional. Anal. i Prilozhen., 1990, vol. 53, pp. 3-5.
49. Krein, M. and Smulian, V., On Regularly Convex Sets in the Space Conjugate to a Banach Space, Ann. of Math. (2), 1940, vol. 41, pp. 556-583.
50. Khurana, S.S., Pointwise Compactness and Measurability, Pacific J. Math., 1979, vol. 83, no. 2, pp. 387-391.
51. Lau, A. T.-M., Finite-Dimensional Invariant Subspaces for a Semigroup of Linear Operators, J. Math. Anal. Appl., 1983, vol. 97, no. 2, pp. 374-379.
52. Lau, A. T.-M., Amenability of Semigroups, The Analytical and Topological Theory of Semigroups, de Gruyter Exp. Math., 1, de Gruyter, Berlin, 1990, pp. 313-334.
53. Lawson, J. D., Joint Continuity in Semitopological Semigroups, Illinois J. Math., 1974, vol. 18, pp. 275-285.
54. Lawson, J. D., Points of Continuity for Semigroup Actions, Trans. Amer. Math. Soc., 1984, vol. 284, no. 1, pp. 183-202.
55. K. de Leeuw and Glicksberg, I., Applications of Almost Periodic Compactifications, Acta Math., 1961, vol. 105, pp. 63-97.
56. K. de Leeuw and Glicksberg, I., The Decomposition of Certain Group Representations, Journal D'Analyse, 1965, vol. 15, pp. 135-192.
57. Levitan, B. M. and Zhikov, V. V., Almost Periodic Functions and Differential Equations, Moskov. Gos. Univ., Moscow, 1978; English transl. in:, Cambridge University Press, Cambridge-New York, 1982.
58. Loomis, L. H., The Spectral Characterization of a Class of Almost Periodic Functions, Journal D'Analyse, 1965, vol. 15, pp. 135-192.
59. Lyubich, Yu. I., Introduction to the Theory of Banach Representations of Groups, Vishcha Shkola, Kharkov, 1985; English transl. in:, Birkhäuser Verlag, Basel-Boston, MA, 1988.
60. Maynard, H., A Geometric Characterization of Banach Spaces Possessing the Radon-Nikodym Property, Trans. Amer. Math. Soc., 1973, vol. 185, pp. 493-500.
61. Megrelishvili, M.G., Operator Topologies and Reflexive Representability, Nuclear Groups and Lie Groups Selected Lectures of the Workshop, Madrid, Spain, September 1999 (E. Martin Peinador, J. Nunez Garsia, eds.), Res. Expo. Math. V. 24, Heldermann, Berlin, 2001, pp. 197-208.
62. Megrelishvili, M.G., Fragmentability and Continuity of Semigroup Actions, Semigroup Forum, 1998, vol. 57, pp. 101-126.
63. Miri, M. R. and Pourabdollah, M. A., Minimally Weakly Almost Periodic Semigroups, Far East J. Math. Sci., 1998, Special Volume, Part II, pp. 191-202.
64. Moore, C. C., Extensions and Low Dimensional Cohomology Theory of Locally Compact Groups. I, II, Trans. Amer. Math. Soc., 1964, vol. 113, pp. 40-63, 64-86.
65. Moore, R. T., Measurable, Continuous and Smooth Vectors for Semigroup and Group Representations, Mem. Amer. Math. Soc., 1968, vol. 78.
66. Naimark, M.A. and Štern [Shtern], A. I., Theory of Group Representations, Springer-Verlag, New York-Heidelberg-Berlin, 1982.
67. Namioka, I., Separate Continuity and Joint Continuity, Pacific. J. Math., 1974, vol. 51, pp. 515-531.
68. Namioka, I., Radon-Nikodym Compact Spaces and Fragmentability, Mathematika, 1987, vol. 34, pp. 258-281.
69. Neeb, K.-H., On a Theorem of S. Banach, J. Lie Theory, 1997, vol. 7, no. 2, pp. 293-300.
70. Neeb, K.-H. and Pickrell, D., Supplements to the Papers Entitled: "On a Theorem of S. Banach" and "The Separable Representations of U(H)", J. Lie Theory, 2000, vol. 10, no. 1, pp. 107-109.
71. Pasynkov, B.A., Almost-Metrizable Topological Groups, Dokl. Akad. Nauk SSSR, 1965, vol. 161, no. 2, pp. 281-284.
72. Pestov, V., Review of [69], Math. Rev. 98i:22003, 1998.
73. Ruppert, W., Compact Semitopological Semigroups: An Intrinsic Theory, Lecture Notes in Math., 1079, 1984, Springer-Verlag, Berlin-New York.
74. Ryll-Nardzewski, C., On Fixed Points of Semigroups of Endomorphisms of Linear Spaces, Collection: Proc. Fifth Berkeley Sympos. Math. Statist, and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif., 1967, pp. 55-61.
75. Sanchis, M., Continuous Functions on Locally Pseudocompact Groups, Topology Appl., 1998, vol. 86, no. 1, pp. 5-23.
76. Schaefer, H. H., Topological Vector Spaces, Second ed., Springer-Verlag, New York-Berlin-Heidelberg, 1999.
77. Shtern, A. I., Compact Semitopological Semigroups and Reflexive Representability of Topological Groups, Russian J. Math. Phys., 1994, vol. 2, no. 1, pp. 131-132.
78. Shtern, A. I., Criteria for Weak and Strong Continuity of Representations of Topological Groups in Banach Spaces, Mat. Sb., 2002, vol. 193, no. 9, pp. 139-156.
79. Shtern, A. I., Continuity of Banach Representations in Terms of Point Variations, Russian J. Math. Phys., 2002, vol. 9, no. 2, pp. 250-252.
80. Thomsen, K., Asymptotic Homomorphisms and Equivariant KK-theory, J. Funct. Anal., 1999, vol. 163, no. 2, pp. 324-343.
81. Thomsen, K., Nonstable K-Theory for Operator Algebras, Internat. J. Math., 1991, vol. 4, no. 3, pp. 245-267.
82. Troallic, J. P., Semigroupes semitopologiques et presque-périodicité, Lecture Notes in Math., 998 (1981), Springer-Verlag, pp. 239-251.
83. Valdivia, M., Some New Results on Weak Compactness, J. Funct. Anal., 1977, vol. 24, no. 1, pp. 1-10.
84. Young, N. J., Periodicity of Functional and Representations of Normed Algebras on Reflexive Spaces, Proc. Edinburgh Math. Soc. (2), 1976/77, vol. 20, no. 2, pp. 99-120.
85. Zhang, C.-Y., Vector-Valued Pseudo Almost Periodic Functions, Czechoslovak Math. J., 1997, vol. 47 (122), no. 3, pp. 385-394.