The connectedness of the group of automorphisms of
Author:
F. Ghahramani
Journal:
Trans. Amer. Math. Soc. 302 (1987), 647659
MSC:
Primary 46J35; Secondary 43A20
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]
John
M. Bachar, William
G. Badé, Philip
C. Curtis Jr., H.
Garth Dales, and Marc
P. Thomas (eds.), Radical Banach algebras and automatic
continuity, Lecture Notes in Mathematics, vol. 975,
SpringerVerlag, BerlinNew York, 1983. MR 697577
(84c:46001)
 [2]
W.
F. Donoghue Jr., The lattice of invariant subspaces of a completely
continuous quasinilpotent 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),
no. 1, 149–161. MR 576191
(81j:43009), http://dx.doi.org/10.1112/jlms/s221.1.149
 [4]
F.
Ghahramani, Isomorphisms between radical weighted convolution
algebras, Proc. Edinburgh Math. Soc. (2) 26 (1983),
no. 3, 343–351. MR 722565
(85h:43002), http://dx.doi.org/10.1017/S0013091500004417
 [5]
Sandy
Grabiner, Weighted convolution algebras as analogues of Banach
algebras of power series, Radical Banach algebras and automatic
continuity (Long Beach, Calif., 1981), Lecture Notes in Math.,
vol. 975, Springer, BerlinNew York, 1983, pp. 282–289. MR 697590
(84g:46073)
 [6]
Sandy
Grabiner, Derivations and automorphisms of Banach algebras of power
series, American Mathematical Society, Providence, R. I., 1974.
Memoirs of the American Mathematical Society, No. 146. MR 0415321
(54 #3410)
 [7]
Frederick
P. Greenleaf, Norm decreasing homomorphisms of group algebras,
Pacific J. Math. 15 (1965), 1187–1219. MR 0194911
(33 #3117)
 [8]
Edwin
Hewitt and Kenneth
A. Ross, Abstract harmonic analysis. Vol. I: Structure of
topological groups. Integration theory, group representations, Die
Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press,
Inc., Publishers, New York; SpringerVerlag,
BerlinGöttingenHeidelberg, 1963. MR 0156915
(28 #158)
 [9]
Einar
Hille, Functional Analysis and
SemiGroups, American Mathematical Society Colloquium
Publications, vol. 31, American Mathematical Society, New York, 1948. MR 0025077
(9,594b)
 [10]
Nicholas
P. Jewell and Allan
M. Sinclair, Epimorphisms and derivations on 𝐿¹(0,1)
are continuous, Bull. London Math. Soc. 8 (1976),
no. 2, 135–139. MR 0402507
(53 #6326)
 [11]
B.
E. Johnson, An introduction to the theory of centralizers,
Proc. London Math. Soc. (3) 14 (1964), 299–320. MR 0159233
(28 #2450)
 [12]
Herbert
Kamowitz and Stephen
Scheinberg, Derivations and automorphisms of
𝐿¹(0,1), Trans. Amer. Math.
Soc. 135 (1969),
415–427. MR 0233210
(38 #1533), http://dx.doi.org/10.1090/S00029947196902332104
 [13]
Stephen
Scheinberg, Automorphisms of commutative Banach algebras,
Problems in analysis (papers dedicated to Salomon Bochner, 1969)
Princeton Univ. Press, Princeton, N.J., 1970, pp. 319–323. MR 0352989
(50 #5475)
 [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.), SpringerVerlag, Berlin and New York, 1983. MR 697577 (84c:46001)
 [2]
 W. F. Donoghue, Jr., The lattice of invariant subspaces of a completely continuous quasinilpotent transformation, Pacific J. Math. 7 (1957), 10311035. MR 0092124 (19:1066f)
 [3]
 F. Ghahramani, Homomorphisms and derivations on weighted convolution algebras, J. London Math. Soc. (2) 21 (1980), 149161. MR 576191 (81j:43009)
 [4]
 , Isomorphisms between radical weighted convolution algebras, Proc. Edinburgh Math. Soc. 26 (1983), 343351. 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.), SpringerVerlag, 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), 11871219. MR 0194911 (33:3117)
 [8]
 E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, SpringerVerlag, Berlin, Heidelberg, New York, 1963. MR 0156915 (28:158)
 [9]
 E. Hille and R. S. Philips, Functional analysis and semigroups, 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), 135139. MR 0402507 (53:6326)
 [11]
 B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. 14 (1964), 299320. MR 0159233 (28:2450)
 [12]
 H. Kamomitz and S. Scheinberg, Derivations and automorphisms of , Trans. Amer. Math. Soc. 135 (1969), 415427. 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:
http://dx.doi.org/10.1090/S00029947198708916396
PII:
S 00029947(1987)08916396
Keywords:
Automorphism,
derivation,
connectedness,
quasinilpotent derivation
Article copyright:
© Copyright 1987
American Mathematical Society
