The connectedness of the group of automorphisms of

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 and an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of and endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if denotes the norm of , , , then the group of automorphisms of topologized by is arcwise connected. It is shown that every automorphism of is of the form , where each is a quasinilpotent derivation. It is shown that the group of principal automorphisms of under operator norm topology is arcwise connected, and every automorphism has the form , where , , and is a derivation, and where denotes the extension by continuity of from a dense subalgebra of to .

**[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**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*, 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