On the theory of topological radicals
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 490-546 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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.
@article{CMFD_2018_64_3_a2,
     author = {E. Kissin and Yu. V. Turovskii and V. S. Shulman},
     title = {On the theory of topological radicals},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {490--546},
     year = {2018},
     volume = {64},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a2/}
}
TY  - JOUR
AU  - E. Kissin
AU  - Yu. V. Turovskii
AU  - V. S. Shulman
TI  - On the theory of topological radicals
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2018
SP  - 490
EP  - 546
VL  - 64
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a2/
LA  - ru
ID  - CMFD_2018_64_3_a2
ER  - 
%0 Journal Article
%A E. Kissin
%A Yu. V. Turovskii
%A V. S. Shulman
%T On the theory of topological radicals
%J Contemporary Mathematics. Fundamental Directions
%D 2018
%P 490-546
%V 64
%N 3
%U http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a2/
%G ru
%F CMFD_2018_64_3_a2
E. Kissin; Yu. V. Turovskii; V. S. Shulman. On the theory of topological radicals. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 490-546. http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a2/

[1] V. A. Andrunakievich, “To the definition of a radical of a ring”, Bull. Acad. Sci. USSR. Ser. Math., 16:3 (1952), 217–224 (in Russian) | MR | Zbl

[2] V. A. Andrunakievich, Yu. M. Ryabukhin, Radicals of Algebras and Structural Theory, Nauka, Moscow, 1979 (in Russian)

[3] E. C. Golod, “On nil algebras and finite-approximable $p$-groups”, Bull. Acad. Sci. USSR. Ser. Math., 28:2 (1964), 273–276 (in Russian) | MR | Zbl

[4] N. Dunford, J. Schwartz, Linear Operators, v. 1, IL, Moscow, 1962, (Russian translation)

[5] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, A. I. Shirshov, Rings Close to Associative, Nauka, Moscow, 1978 (in Russian) | MR

[6] E. I. Zel'manov, “On Engel Lie-algebras”, Rep. Acad. Sci. USSR, 292:2 (1987), 265–268 (in Russian) | MR | Zbl

[7] A. G. Kurosh, “Radicals of rings and algebras”, Math. Digest, 33(75):1 (1953), 13–26 (in Russian) | MR | Zbl

[8] V. I. Lomonosov, “Invariant subspaces for the family of operators which commute with a completely continuous operator”, Funct. Anal. Appl., 7:3 (1973), 55–56 (in Russian) | MR | Zbl

[9] Yu. V. Turovskii, “On spectral properties of some Lie-subalgebras and spectral radius of subsets in Banach algebras”, Spectr. Theor. Oper. Appl., 1985, no. 6, 144–181 (in Russian)

[10] Yu. V. Turovskii, B. C. Shul'man, “Radicals in Banach algebras and some problems of the theory of radical Banach algebras”, Funct. Anal. Appl., 35:4 (2001), 88–91 (in Russian) | DOI | MR | Zbl

[11] Yu. V. Turovskii, B. C. Shul'man, “Topological radicals and mutual spectral radius”, Funct. Anal. Appl., 46:4 (2012), 61–82 (in Russian) | DOI | MR | Zbl

[12] B. C. Shul'man, “On invariant subspaces of Volterra operators”, Funct. Anal. Appl., 18:2 (1984), 85–86 (in Russian) | MR | Zbl

[13] Albert A. A., “The radical of a non-associative algebra”, Bull. Am. Math. Soc., 48 (1942), 891–897 | DOI | MR | Zbl

[14] Alexander J. C., “Compact Banach algebras”, Proc. London Math. Soc. (3), 18 (1968), 1–18 | DOI | MR | Zbl

[15] Amitsur S. A., “A general theory of radicals, I: Radicals in complete lattices”, Amer. J. Math., 74 (1952), 774–786 | DOI | MR | Zbl

[16] Amitsur S. A., “A general theory of radicals, II: Rings and bicategories”, Amer. J. Math., 76:1 (1954), 100–125 | DOI | MR | Zbl

[17] Amitsur S. A., “A general theory of radicals, III: Applications”, Amer. J. Math., 76:1 (1954), 126–136 | DOI | MR | Zbl

[18] Andreolas G., Anoussis M., Topological radicals of nest algebras, 10 Oct. 2016, arXiv: 1608.05857v2[math.OA] | MR

[19] Argiros S. A., Haydon R., “A hereditarily indecomposable $L_\infty$-space that solves the scalar-plus-compact problem”, Acta Math., 206 (2011), 1–54 | DOI | MR

[20] Aupetit B., Propriétés spectrales des algèbres de Banach, Springer, Berlin, 1979 | MR

[21] Aupetit B., Primer to spectral theory, Springer, N.Y., 1991 | MR

[22] Aupetit B., Mathieu M., “The continuity of Lie homomorphisms”, Stud. Math., 138 (2000), 193–199 | MR | Zbl

[23] Baer R., “Radical ideals”, Amer. J. Math., 65 (1943), 537–568 | DOI | MR | Zbl

[24] Barnes B. A., Murphy G. J., Smyth M. R. F., West T. T., Riesz and Fredholm theory in Banach algebras, Pitman Publ. Inc., Boston, 1982 | MR | Zbl

[25] Berger M. A., Wang Y., “Bounded semigroups of matrices”, Linear Algebra Appl., 166 (1992), 21–27 | DOI | MR | Zbl

[26] Bonsall F. F., “Operators that act compactly on an algebra of operators”, Bull. London Math. Soc., 1 (1969), 163–170 | DOI | MR | Zbl

[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, 58–128 | DOI | MR | Zbl

[28] Brown F., McCoy N. H., “Some theorems on groups with applications to ring theory”, Trans. Am. Math. Soc., 69 (1950), 302–311 | DOI | MR | Zbl

[29] Burlando L., “Spectral continuity in some Banach algebras”, Rocky Mountain J. Math., 23 (1993), 17–39 | DOI | MR | Zbl

[30] Curto R. E., “Spectral theory of elementary operators”, Elementary operators and applications, World Sci. Publ., Singapour–New Jersey–London, 1992, 3–54 | MR

[31] Davidson K. R., $C^*$-algebras by examples, Am. Math. Soc., Providence, 1996 | MR | Zbl

[32] Defant A., Floret K., Tensor norms and operator ideals, Elsevier, Amsterdam, 1993 | MR

[33] Divinsky N. J., Rings and radicals, Allen and Unwin, London, 1965 | MR | Zbl

[34] Dixmier J., Les $C^*$-algébres et leur reprèsentations, Gauthier-Villars, Paris, 1964 | MR | Zbl

[35] Dixon P. G., “A Jacobson-semisimple Banach algebra with a dense nil subalgebra”, Colloq. Math., 37 (1977), 81–82 | DOI | MR | Zbl

[36] Dixon P. G., “Topologically nilpotent Banach algebras and factorization”, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 329–341 | DOI | MR | Zbl

[37] Dixon P. G., “Topologically irreducible representations and radicals in Banach algebras”, Proc. London Math. Soc. (3), 74 (1997), 174–200 | DOI | MR | Zbl

[38] Dixon P. G., Müller V., “A note on topologically nilpotent Banach algebras”, Stud. Math., 102 (1992), 269–275 | DOI | MR | Zbl

[39] Dixon P. G., Willis G. A., “Approximate identities in extensions of topologically nilpotent Banach algebras”, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 45–52 | DOI | MR | Zbl

[40] Feldman I., Krupnik N., “On the continuity of the spectrum in certain Banach algebras”, Integral Equ. Operator Theory, 38 (2000), 284–301 | DOI | MR | Zbl

[41] Gardner B. J., Wieland R., Radical theory of rings, Marcel Dekker Inc., New York, 2004 | MR | Zbl

[42] Gray M., A radical approach to algebra, Addison-Wesley Publ. Comp., Massachusetts, 1970 | MR | Zbl

[43] Guinand P. G., “On quasinilpotent semigroups of operators”, Proc. Am. Math. Soc., 86 (1982), 485–486 | DOI | MR | Zbl

[44] Halmos P., Hilbert space problem book, Van Nostrand, Toronto–London, 1967 | MR | Zbl

[45] Hayman W. K., Kennedy C. B., Subharmonic functions, v. 1, Academic Press, London–New York–San Francisko, 1976 | MR | Zbl

[46] Jacobson N., “The radical and semi simplicity for arbitrary rings”, Am. J. Math., 67 (1945), 300–320 | DOI | MR | Zbl

[47] Jungers R., Joint spectral radius, theory and applications, Springer, Berlin, 2009 | MR

[48] Kennedy M., Shulman V. S., Turovskii Yu. V., “Invariant subspaces of subgraded Lie algebras of compact operators”, Integral Equ. Operator Theory, 63 (2009), 47–93 | DOI | MR | Zbl

[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), 80 (2009), 603–626 | DOI | MR | Zbl

[50] Kissin E., Shulman V. S., Turovskii Yu. V., “Topological radicals and Frattini theory of Banach Lie algebras”, Integral Equ. Operator Theory, 74 (2012), 51–121 | DOI | MR | Zbl

[51] Köthe G., “Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollstandig reduzibel ist”, Math. Z., 32 (1930), 161–186 | DOI | MR | Zbl

[52] Köthe G., Topological vector spaces, v. I, Spinger, New York, 1969 | MR

[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, 63:2 (2010), 417–424 | MR | Zbl

[55] Lebow A., Schechter M., “Semigroups of operators and measures of noncompactness”, J. Funct. Anal., 7 (1971), 1–26 | DOI | MR | Zbl

[56] Levitzki A., “On the radical of a general ring”, Bull. Am. Math. Soc., 43 (1943), 462–466 | DOI | MR

[57] Morris I. D., The generalized Berger–Wang formula and the spectral radius of linear cocycles, Preprint, 16 Jun. 2009, arXiv: 0906.2915v1[math.DS] | MR

[58] Newburgh J. D., “The variation of spectra”, Duke Math. J., 18 (1951), 165–176 | DOI | MR | Zbl

[59] Palacios A. R., “The uniqueness of the complete norm topology in complete normed nonassociative algebras”, J. Funct. Anal., 60 (1985), 1–15 | DOI | MR | Zbl

[60] Peng C., Turovskii Yu., “Topological radicals, VI. Scattered elements in Banach, Jordan, and associative algebras”, Stud. Math., 235 (2016), 171–208 | MR | Zbl

[61] Peters J. R., Wogen R. W., “Commutative radical operator algebras”, J. Operator Theory, 42 (1999), 405–424 | MR | Zbl

[62] Pietsch A., Operator ideals, Veb Deutscher Verlag der Wissenschaften, Berlin, 1978 | MR | Zbl

[63] Pietsch A., History of Banach spaces and linear operators, Birkhauser, Boston, 2007 | MR | Zbl

[64] Protasov V. Yu., “The generalized joint spectral radius. A geometric approach”, Izv. Math., 61:5 (1997), 995–1030 | DOI | MR | Zbl

[65] Radjavi H., Rosenthal P., Simultaneous triangularization, Springer, N.Y., 2000 | MR | Zbl

[66] Read C. J., “Quasinilpotent operators and the invariant subspace problem”, J. London Math. Soc. (2), 56 (1997), 595–606 | DOI | MR | Zbl

[67] Ringrose J. R., “On some algebras of operators”, Proc. London Math. Soc., 15 (1965), 61–83 | DOI | MR | Zbl

[68] Rota G.-C., Strang W. G., “A note on the joint spectral radius”, Indag. Math., 22 (1960), 379–381 | DOI | MR

[69] Shulman T., Continuity of spectral radius and type I $C^*$-algebras, arXiv: ; Proc. Am. Math. Soc., 147:2 (2019), 641–646 1707.08848 | MR | Zbl

[70] Shulman V. S., Turovskii Yu. V., “Joint spectral radius, operator semigroups and a problem of a Wojtynski”, J. Funct. Anal., 177 (2000), 383–441 | DOI | MR | Zbl

[71] Shulman V. S., Turovskii Yu. V., “Formulae for joint spectral radii of sets of operators”, Stud. Math., 149 (2002), 23–37 | DOI | MR | Zbl

[72] Shulman V. S., Turovskii Yu. V., “Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action”, J. Funct. Anal., 223 (2005), 425–508 | DOI | MR | Zbl

[73] Shulman V. S., Turovskii Yu. V., “Topological radicals, I. Basic properties, tensor products and joint quasinilpotence”, Banach Center Publ., 67 (2005), 293–333 | DOI | MR | Zbl

[74] Shulman V. S., Turovskii Yu. V., “Topological radicals, II. Applications to the spectral theory of multiplication operators”, Oper. Theory Adv. Appl., 212 (2010), 45–114 | MR

[75] Shulman V. S., Turovskii Yu. V., “Topological radicals, V. From algebra to spectral theory”, Oper. Theory Adv. Appl., 233 (2014), 171–280 | DOI | MR | Zbl

[76] Szász F. A., Radicals of rings, Akadémiai Kiadó, Budapest, 1981

[77] Turovskii Yu. V., “Volterra semigroups have invariant subspaces”, J. Funct. Anal., 182 (1999), 313–323 | DOI | MR

[78] Vala K., “On compact sets of compact operators”, Ann. Acad. Sci. Fenn. Math., 351 (1964), 1–8 | MR

[79] Vesentini E., “On the subharmonicity of the spectral radius”, Boll. Unione Mat. Ital. (9), 4 (1968), 427–429 | MR

[80] Willis G., “Compact approximation property does not imply approximation property”, Stud. Math., 103 (1992), 99–108 | DOI | MR | Zbl

[81] Wojtynski W., “A note on compact Banach–Lie algebras of Volterra type”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys., 26:2 (1978), 105–107 | MR | Zbl

[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., 26:2 (1978), 109–113 | MR | Zbl

[83] Wojtynski W., “Quasinilpotent Banach–Lie algebras are Baker–Campbell–Hausdorff”, J. Funct. Anal., 153 (1998), 405–413 | DOI | MR | Zbl

[84] Zemanek J., “Spectral characterization of two-sided ideals in Banach algebras”, Stud. Math., 67 (1980), 1–12 | DOI | MR | Zbl