On the Theory of Topological Radicals

Cover Page

Cite item

Full Text

Abstract

In this paper, we review main directions and results of the theory of topological radicals. We consider applications to different problems in the theory of operators and Banach algebras.

About the authors

E V Kissin

London Metropolitan University

Author for correspondence.
Email: e.kissin@londonmet.ac.uk
166-220, Holloway Road, N7 8DB, UK

Yu V Turovskii

Email: yuri.turovskii@gmail.com

V S Shulman

Vologda State University

Email: victor.shulman80@gmail.com
Vologda, Russia

References

  1. Андрунакиевич В. A. K определению радикала кольца// Изв. АН СССР. Сер. Мат. - 1952. - 16.- C. 217-224.
  2. Андрунакиевич В. А., Рябухин Ю. М. Радикалы алгебр и структурная теория. - М.: Наука, 1979.
  3. Голод E. C. О нильалгебрах и финитно-аппроксимируемых p-группах// Изв. АН СССР. Сер. Мат. - 1964. - 28. - C. 273-276.
  4. Данфорд Н., Шварц Дж. Линейные операторы. T. 1. - М.: ИЛ, 1962.
  5. Жевлаков K. A., Слинько A. M., Шестаков И. П., Ширшов А. И. Кольца, близкие к ассоциативным. - М.: Наука, 1978.
  6. Зельманов E. И. Об энгелевых алгебрах Ли// Докл. АН СССР. - 1987. - 292, № 2. - C. 265-268.
  7. Курош А. Г. Радикалы колец и алгебр// Мат. сб. - 1953. - 33, № 1. - C. 13-26.
  8. Ломоносов В. И. Инвариантные подпространства для операторов, коммутирующих с компактными операторами// Функц. анализ и его прилож. - 1973. - 7. - C. 213-214.
  9. Туровский Ю. В. О спектральных свойствах некоторых лиевых подалгебр и спектральном радиусе подмножеств в банаховых алгебрах// Спектр. теор. опер. и ее прилож. - 1985. - 6. - C. 144-181.
  10. Туровский Ю. В., Шульман B. C. Радикалы в банаховых алгебрах и некоторые проблемы теории радикальных банаховых алгебр// Функц. анализ и его прилож. - 2001. - 35, № 4. - C. 88-91.
  11. Туровский Ю. В., Шульман B. C. Топологические радикалы и совместный спектральный радиус// Функц. анализ и его прилож. - 2012. - 46, № 4. - C. 61-82.
  12. Шульман B. C. Об инвариантных подпространствах вольтерровых операторов// Функц. анализ и его прилож. - 1984. - 18, № 2. - C. 84-85.
  13. Albert A. A. The radical of a non-associative algebra// Bull. Am. Math. Soc. - 1942. - 48. - С. 891-897.
  14. Alexander J. C. Compact Banach algebras// Proc. London Math. Soc. (3). - 1968. - 18. - С. 1-18.
  15. Amitsur S. A. A general theory of radicals, I: Radicals in complete lattices// Amer. J. Math. - 1952. - 74. - С. 774-786.
  16. Amitsur S. A. A general theory of radicals, II: Rings and bicategories// Amer. J. Math. - 1954. - 76, № 1. - С. 100-125.
  17. Amitsur S. A. A general theory of radicals, III: Applications// Amer. J. Math. - 1954. - 76, № 1. - С. 126-136.
  18. Andreolas G., Anoussis M. Topological radicals of nest algebras// arXiv:1608.05857v2 [math.OA] 10 Oct 2016.
  19. Argiros S. A., Haydon R. A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem// Acta Math. - 2011. - 206.- С. 1-54.
  20. Aupetit B. Proprie´te´s spectrales des alge`bres de Banach. - Berlin: Springer, 1979.
  21. Aupetit B. Primer to spectral theory. - N.Y.: Springer, 1991.
  22. Aupetit B., Mathieu M. The continuity of Lie homomorphisms// Stud. Math. - 2000. - 138. - С. 193- 199.
  23. Baer R. Radical ideals// Amer. J. Math. - 1943. - 65. - С. 537-568.
  24. Barnes B. A., Murphy G. J., Smyth M. R. F., West T. T. Riesz and Fredholm theory in Banach algebras. - Boston: Pitman Publ. Inc., 1982.
  25. Berger M. A., Wang Y. Bounded semigroups of matrices// Linear Algebra Appl. - 1992. - 166. - С. 21- 27.
  26. Bonsall F. F. Operators that act compactly on an algebra of operators// Bull. London Math. Soc. - 1969. - 1. - С. 163-170.
  27. Brown L. G., Douglas R. G., Fillmore P. A. Unitary equivalence modulo the compact operators and extensions of C*-algebras// Proc. of Conf. on Operator Theory, Halifax, Nova Scotia. - 1973. - С. 58-128.
  28. Brown F., McCoy N. H. Some theorems on groups with applications to ring theory// Trans. Am. Math. Soc. - 1950. - 69. - С. 302-311
  29. Burlando L. Spectral continuity in some Banach algebras// Rocky Mountain J. Math. - 1993. - 23.- С. 17-39.
  30. Curto R. E. Spectral theory of elementary operators// В сб.: «Elementary operators and applications». - Singapour-New Jersey-London: World Sci. Publ., 1992. - С. 3-54.
  31. Davidson K. R. C*-algebras by examples. - Providence: Am. Math. Soc., 1996.
  32. Defant A., Floret K. Tensor norms and operator ideals. - Amsterdam: Elsevier, 1993.
  33. Divinsky N. J. Rings and radicals. - London: Allen and Unwin, 1965.
  34. Dixmier J. Les C*-alge´bres et leur repre`sentations. - Paris: Gauthier-Villars, 1964.
  35. Dixon P. G. A Jacobson-semisimple Banach algebra with a dense nil subalgebra// Colloq. Math. - 1977. - 37. - С. 81-82.
  36. Dixon P. G. Topologically nilpotent Banach algebras and factorization// Proc. Roy. Soc. Edinburgh Sect. A. - 1991. - 119. - С. 329-341.
  37. Dixon P. G. Topologically irreducible representations and radicals in Banach algebras// Proc. London Math. Soc. (3). - 1997. - 74. - С. 174-200.
  38. Dixon P. G., Mu¨ ller V. A note on topologically nilpotent Banach algebras// Stud. Math. - 1992. - 102.- С. 269-275.
  39. Dixon P. G., Willis G. A. Approximate identities in extensions of topologically nilpotent Banach algebras// Proc. Roy. Soc. Edinburgh Sect. A. - 1992. - 122. - С. 45-52.
  40. Feldman I., Krupnik N. On the continuity of the spectrum in certain Banach algebras// Integral Equ. Operator Theory. - 2000. - 38. - С. 284-301.
  41. Gardner B. J., Wieland R. Radical theory of rings. - New York: Marcel Dekker Inc., 2004.
  42. Gray M. A radical approach to algebra. - Massachusetts: Addison-Wesley Publ. Comp., 1970.
  43. Guinand P. G. On quasinilpotent semigroups of operators// Proc. Am. Math. Soc. - 1982. - 86. - С. 485- 486.
  44. Halmos P. Hilbert space problem book. - Toronto-London: Van Nostrand, 1967.
  45. Hayman W. K., Kennedy С. B. Subharmonic functions. Vol. 1. - London-New York-San Francisko: Academic Press, 1976.
  46. Jacobson N. The radical and semi simplicity for arbitrary rings// Am. J. Math. - 1945. - 67. - С. 300- 320.
  47. Jungers R. Joint spectral radius, theory and applications. - Berlin: Springer, 2009.
  48. Kennedy M., Shulman V. S., Turovskii Yu. V. Invariant subspaces of subgraded Lie algebras of compact operators// Integral Equ. Operator Theory. - 2009. - 63. - С. 47-93.
  49. Kissin E., Shulman V. S., Turovskii Yu. V. Banach Lie algebras with Lie subalgebras of finite codimension have Lie ideals// J. London Math. Soc. (2). - 2009. - 80. - С. 603-626.
  50. Kissin E., Shulman V. S., Turovskii Yu. V. Topological radicals and Frattini theory of Banach Lie algebras// Integral Equ. Operator Theory. - 2012. - 74. - С. 51-121
  51. Ko¨the G. Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollstandig reduzibel ist// Math. Z. - 1930. - 32. - С. 161-186.
  52. Ko¨the G. Topological vector spaces. I. - New York: Spinger, 1969.
  53. Kozyakin V. An annotated bibliography on convergence of matrix products and the theory of convergence of the joint/generalized spectral radius// Preprint Inst. Inform. Transmission Prob., 2013.
  54. Kusuda M. A characterization of scattered C*-algebras and its applications to C*-crossed products// J. Operator Theory. - 2010. - 63, № 2. - С. 417-424.
  55. Lebow A., Schechter M. Semigroups of operators and measures of noncompactness// J. Funct. Anal. - 1971. - 7.- С. 1-26.
  56. Levitzki A. On the radical of a general ring// Bull. Am. Math. Soc. - 1943. - 43. - С. 462-466.
  57. Morris I. D. The generalized Berger-Wang formula and the spectral radius of linear cocycles// Preprint. - ArXiv:0906.2915v1 [math.DS] 16 Jun 2009.
  58. Newburgh J. D. The variation of spectra// Duke Math. J. - 1951. - 18. - С. 165-176.
  59. Palacios A. R. The uniqueness of the complete norm topology in complete normed nonassociative algebras// J. Funct. Anal. - 1985. - 60.- С. 1-15.
  60. Peng C., Turovskii Yu. Topological radicals, VI. Scattered elements in Banach, Jordan, and associative algebras// Stud. Math. - 2016. - 235. - С. 171-208.
  61. Peters J. R., Wogen R. W. Commutative radical operator algebras// J. Operator Theory. - 1999. - 42.- С. 405-424.
  62. Pietsch A. Operator ideals. - Berlin: Veb Deutscher Verlag der Wissenschaften, 1978.
  63. Pietsch A. History of Banach spaces and linear operators. - Boston: Birkhauser, 2007.
  64. Protasov V. Yu. The generalized joint spectral radius. A geometric approach// Izv. Math. - 1997. - 61, № 5. - С. 995-1030.
  65. Radjavi H., Rosenthal P. Simultaneous triangularization. - N.Y.: Springer, 2000.
  66. Read C. J. Quasinilpotent operators and the invariant subspace problem// J. London Math. Soc. (2). - 1997. - 56. - С. 595-606.
  67. Ringrose J. R. On some algebras of operators // Proc. London Math. Soc. - 1965. - 15. - С. 61-83.
  68. Rota G.-C., Strang W. G. A note on the joint spectral radius// Indag. Math. - 1960. - 22. - С. 379-381.
  69. Shulman T. Continuity of spectral radius and type I C*-algebras// arXiv: 1707.08848 (to appear in Proc. Am. Math. Soc.).
  70. Shulman V. S., Turovskii Yu. V. Joint spectral radius, operator semigroups and a problem of a Wojtynski// J. Funct. Anal. - 2000. - 177. - С. 383-441.
  71. Shulman V. S., Turovskii Yu. V. Formulae for joint spectral radii of sets of operators// Stud. Math. - 2002. - 149. - С. 23-37.
  72. Shulman V. S., Turovskii Yu. V. Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action// J. Funct. Anal. - 2005. - 223. - С. 425-508.
  73. Shulman V. S., Turovskii Yu. V. Topological radicals, I. Basic properties, tensor products and joint quasinilpotence// Banach Center Publ. - 2005. - 67. - С. 293-333.
  74. Shulman V. S., Turovskii Yu. V. Topological radicals, II. Applications to the spectral theory of multiplication operators// Oper. Theory Adv. Appl. - 2010. - 212. - С. 45-114.
  75. Shulman V. S., Turovskii Yu. V. Topological radicals, V. From algebra to spectral theory// Oper. Theory Adv. Appl. - 2014. - 233. - С. 171-280.
  76. Sza´ sz F. A. Radicals of rings - Budapest: Akade´miai Kiado´, 1981.
  77. Turovskii Yu. V. Volterra semigroups have invariant subspaces// J. Funct. Anal. - 1999. - 182. - С. 313- 323.
  78. Vala K. On compact sets of compact operators// Ann. Acad. Sci. Fenn. Math. - 1964. - 351. - С. 1-8.
  79. Vesentini E. On the subharmonicity of the spectral radius// Boll. Unione Mat. Ital. (9). - 1968. - 4.- С. 427-429.
  80. Willis G. Compact approximation property does not imply approximation property// Stud. Math. - 1992. - 103. - С. 99-108.
  81. Wojtynski W. A note on compact Banach-Lie algebras of Volterra type// Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. - 1978. - 26, № 2. - С. 105-107.
  82. Wojtynski W. On the existence of closed two-sided ideals in radical Banach algebras with compact elements// Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. - 1978. - 26, № 2. - С. 109-113.
  83. Wojtynski W. Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff// J. Funct. Anal. - 1998. - 153. - С. 405-413.
  84. Zemanek J. Spectral characterization of two-sided ideals in Banach algebras// Stud. Math. - 1980. - 67.- С. 1-12.

Copyright (c) 2019 Contemporary Mathematics. Fundamental Directions

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies