Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2227310.22363/2413-3639-2018-64-3-490-546New ResultsНовые результатыResearch ArticleOn the Theory of Topological RadicalsО теории топологических радикаловKissinE VКиссинЭ Вe.kissin@londonmet.ac.ukTurovskiiYu VТуровскийЮ Вyuri.turovskii@gmail.comShulmanV SШульманВ Сvictor.shulman80@gmail.comLondon Metropolitan UniversityVologda State UniversityВологодский государственный университет15122018643Proceedings of the Crimean Autumn Mathematical School-SymposiumТруды Крымской осенней математической школы-симпозиума49054629112019Copyright ©; 2019, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2019, Современная математика. Фундаментальные направления2019Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://journals.rudn.ru/CMFD/article/view/22273

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.

В работе обсуждаются основные направления и результаты теории топологических радикалов. Рассматриваются приложения к различным проблемам теории операторов и теории банаховых алгебр.

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