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1994 | 30 | 1 | 259-265
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Nil, nilpotent and PI-algebras

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The notions of nil, nilpotent or PI-rings (= rings satisfying a polynomial identity) play an important role in ring theory (see e.g. [8], [11], [20]). Banach algebras with these properties have been studied considerably less and the existing results are scattered in the literature. The only exception is the work of Krupnik [13], where the Gelfand theory of Banach PI-algebras is presented. However, even this work has not get so much attention as it deserves.
The present paper is an attempt to give a survey of results concerning Banach nil, nilpotent and PI-algebras.
The author would like to thank to J. Zemánek for essential completion of the bibliography.
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  • Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
  • [1] P. G. Dixon, Locally finite Banach algebras, J. London Math. Soc. 8 (1974), 325-328.
  • [2] P. G. Dixon, Topologically nilpotent Banach algebras and factorization, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), 329-341.
  • [3] P. G. Dixon and V. Müller, A note on topologically nilpotent Banach algebras, Studia Math. 102 (1992), 269-275.
  • [4] J. Dubnov et V. Ivanov, Sur l'abaissement du degré des polynômes en affineurs, C. R. (Doklady) Acad. Sci. URSS 41 (1943), 95-98.
  • [5] J. Duncan and A. W. Tullo, Finite dimensionality, nilpotents and quasinilpotents in Banach algebras, Proc. Edinburgh Math. Soc. 19 (1974/75), 45-49.
  • [6] E. Formanek, The Nagata-Higman Theorem, Acta Appl. Math. 21 (1990), 185-192.
  • [7] S. Grabiner, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc. 21 (1969), 510.
  • [8] I. N. Herstein, Noncommutative Rings, Carus Math. Monographs 15, Math. Assoc. Amer., Wiley, 1968.
  • [9] G. Higman, On a conjecture of Nagata, Proc. Cambridge Philos. Soc. 52 (1956), 1-4.
  • [10] R. A. Hirschfeld and B. E. Johnson, Spectral characterization of finite-dimensional algebras, Indag. Math. 34 (1972), 19-23.
  • [11] N. Jacobson, Structure of Rings, third edition, Amer. Math. Soc. Colloq. Publ. 37, Amer. Math. Soc., Providence, R.I., 1968.
  • [12] I. Kaplansky, Ring isomorphisms of Banach algebras, Canad. J. Math. 6 (1954), 374-381.
  • [13] N. Ya. Krupnik, Banach Algebras with Symbol and Singular Integral Operators, Birkhäuser, Basel, 1987.
  • [14] E. N. Kuzmin, On the Nagata-Higman Theorem, in: Mathematical Structures-Computational Mathematics-Mathematical Modeling, Proceedings dedicated to the sixtieth birthday of Academician L. Iliev, Sofia, 1975, 101-107 (in Russian).
  • [15] V. Müller, Kaplansky's theorem and Banach PI-algebras, Pacific J. Math. 141 (1990), 355-361.
  • [16] M. Nagata, On the nilpotency of nil-algebras, J. Math. Soc. Japan 4 (1952), 296-301.
  • [17] K. M. Przyłuski and S. Rolewicz, On stability of linear time varying infinite-dimensional discrete-time systems, Systems Control Lett. 4 (1984), 307-315.
  • [18] Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723-756 (in Russian).
  • [19] G. C. Rota and W. G. Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), 379-381.
  • [20] L. H. Rowen, Polynomial Identities in Ring Theory, Academic Press, New York, 1980.
  • [21] V. S. Shul'man, On invariant subspaces of Volterra operators, Funct. Anal. Appl. 18 (1984), 85-86.
  • [22] Yu. V. Turovskiĭ, Spectral properties of certain Lie subalgebras and the spectral radius of subsets of a Banach algebra, in: Spectral Theory of Operators and its Applications, No. 6, Elm, Baku, 1985, 144-181 (in Russian).
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