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1992 | 103 | 1 | 51-69
Tytuł artykułu

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

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EN
We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L¹_{loc}$ of locally integrable functions on the half-line $ℝ^+$. We show, among other things, that every automorphism θ of $L¹_{loc}$ is of the form $θ = φ _ae^{λX}e^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_a$ is the dilation operator $(φ_af)(x) = af(ax)$ ($f ∈ L¹_{loc}$, $x ∈ ℝ^+$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded sets and is the semidirect product of a connected subgroup and a discrete group which is isomorphic to the discrete group of real numbers.
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Twórcy
  • Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
  • epartment of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Bibliografia
  • [1] W. G. Bade and H. G. Dales, Norms and ideals in radical convolution algebras, J. Funct. Anal. 41 (1) (1981), 77-109.
  • [2] H. G. Dales, Convolution algebras on the real line, in: Radical Banach Algebras and Automatic Continuity, J. M. Bachar et al. (eds.), Lecture Notes in Math. 975, Springer, Berlin 1983, 180-209.
  • [3] H. G. Dales, Positive Results in Automatic Continuity, forthcoming book.
  • [4] H. G. Diamond, Characterization of derivations on an algebra of measures, Math. Z. 100 (1967), 135-140.
  • [5] H. G. Diamond, Characterization of derivations on an algebra of measures II, ibid. 105 (1968), 301-306.
  • [6] W. F. Donoghue, Jr., The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math. 7 (1957), 1031-1035.
  • [7] I. Gelfand, D. Raikov and G. Shilov, Commutative Normed Rings, Chelsea, 1964.
  • [8] F. Ghahramani, Homomophisms and derivations on weighted convolution algebras, J. London Math. Soc. (2) 21 (1980), 149-161.
  • [9] F. Ghahramani, Isomorphisms between radical weighted convolution algebras, Proc. Edinburgh Math. Soc. 26 (1983), 343-351.
  • [10] F. Ghahramani, The connectedness of the group of automorphisms of L¹(0,1), Trans. Amer. Math. Soc. 302 (2) (1987), 647-659.
  • [11] F. Ghahramani, The group of automorphisms of L¹(0,1) is connected, ibid. 314 (2) (1989), 851-859.
  • [12] F. Ghahramani and J. P. McClure, Automorphisms of radical weighted convolution algebras, J. London Math. Soc. (2) 41 (1990), 122-132.
  • [13] S. Grabiner, Derivations and automorphisms of Banach algebras of power series, Mem. Amer. Math. Soc. 146 (1974).
  • [14] N. P. Jewell and A. M. Sinclair, Epimorphisms and derivations on L¹(0,1) are continuous, Bull. London Math. Soc. 8 (1976), 135-139.
  • [15] H. Kamowitz and S. Scheinberg, Derivations and automorphisms of L¹(0,1), Trans. Amer. Math. Soc. 135 (1969), 415-427.
  • [16] E. A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952).
  • [17] A. P. Robertson and W. J. Robertson, Topological Vector Spaces, Cambridge Univ. Press, London 1964.
  • [18] M. P. Thomas, Approximation in the radical algebra ℓ¹(w) when ${w_n}$ is star-shaped, in: Radical Banach Algebras and Automatic Continuity, J. M. Bachar et al. (eds.), Lecture Notes in Math. 975, Springer, Berlin 1983, 258-272.
  • [19] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967.
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Bibliografia
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