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The connectedness of the group of automorphisms of $ L\sp 1(0,1)$


Author: F. Ghahramani
Journal: Trans. Amer. Math. Soc. 302 (1987), 647-659
MSC: Primary 46J35; Secondary 43A20
DOI: https://doi.org/10.1090/S0002-9947-1987-0891639-6
MathSciNet review: 891639
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Abstract: For each of the radical Banach algebras $ {L^1}(0,1)$ and $ {L^1}(w)$ an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of $ {L^1}(0,1)$ and $ {L^1}(w)$ endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ denotes the norm of $ B({L^p}(0,1)$, $ {L^1}(0,1))$, $ 1 < p \leq \infty $, then the group of automorphisms of $ {L^1}(0,1)$ topologized by $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ is arcwise connected. It is shown that every automorphism $ \theta $ of $ {L^1}(0,1)$ is of the form $ \theta = {e^{\lambda d}}{\operatorname{lim}}{e^{qn}}({\text{BSO}})$, where each $ {q_n}$ is a quasinilpotent derivation. It is shown that the group of principal automorphisms of $ {l^1}(w)$ under operator norm topology is arcwise connected, and every automorphism has the form $ {e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$, where $ \alpha \in {\mathbf{R}}$, $ \lambda > 0$, and $ D$ is a derivation, and where $ {({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$ denotes the extension by continuity of $ {e^{\lambda d}}{e^D}{e^{ - \lambda d}}$ from a dense subalgebra of $ {l^1}(w)$ to $ {l^1}(w)$.


References [Enhancements On Off] (What's this?)

  • [1] H. G. Dales, Convolution algebras on the real line, Radical Banach algebras and automatic continuity, Proc. Long Beach 1981 (J. M. Bachar et al., eds.), Springer-Verlag, Berlin and New York, 1983. MR 697577 (84c:46001)
  • [2] W. F. Donoghue, Jr., The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math. 7 (1957), 1031-1035. MR 0092124 (19:1066f)
  • [3] F. Ghahramani, Homomorphisms and derivations on weighted convolution algebras, J. London Math. Soc. (2) 21 (1980), 149-161. MR 576191 (81j:43009)
  • [4] -, Isomorphisms between radical weighted convolution algebras, Proc. Edinburgh Math. Soc. 26 (1983), 343-351. MR 722565 (85h:43002)
  • [5] S. Grabiner, Weighted convolution algebras as analogues of Banach algebras of power series, Radical Banach Algebras and Automatic Continuity, Proc. Long Beach 1981 (J. M. Bachar et al. eds.), Springer-Verlag, Berlin and New York, 1983. MR 697590 (84g:46073)
  • [6] -, Derivations and automorphisms of Banach algebras of power series, Mem. Amer. Math. Soc. No. 146, 1974. MR 0415321 (54:3410)
  • [7] F. P. Greenleaf, Normdecreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187-1219. MR 0194911 (33:3117)
  • [8] E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, Springer-Verlag, Berlin, Heidelberg, New York, 1963. MR 0156915 (28:158)
  • [9] E. Hille and R. S. Philips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, rev. ed., Providence, R.I., 1957. MR 0025077 (9:594b)
  • [10] N. P. Jewell and A. M. Sinclair, Epimorphisms and derivations on $ {L^1}(0,1)$ are continuous, Bull. London Math. Soc. 8 (1976), 135-139. MR 0402507 (53:6326)
  • [11] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. 14 (1964), 299-320. MR 0159233 (28:2450)
  • [12] H. Kamomitz and S. Scheinberg, Derivations and automorphisms of $ {L^1}(0,1)$, Trans. Amer. Math. Soc. 135 (1969), 415-427. MR 0233210 (38:1533)
  • [13] S. Scheinberg, Automorphims of commutative Banach algebras, Problems in Analysis, A Symposium in Honor of Salomon Bochner (R. C. Gunning, ed.), Princeton Univ. Press, Princeton, N. J., 1970. MR 0352989 (50:5475)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0891639-6
Keywords: Automorphism, derivation, connectedness, quasinilpotent derivation
Article copyright: © Copyright 1987 American Mathematical Society

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