Let A be a commutative unital Banach algebra and let A/ℛ be the quotient algebra of A modulo its radical ℛ. This paper is concerned with raising bounded groups in A/ℛ to bounded groups in the algebra A. The results will be applied to the problem of splitting radical extensions of certain Banach algebras.
@article{bwmeta1.element.bwnjournal-article-smv141i1p85bwm,
author = {W. Bade and P. Curtis and A. Sinclair},
title = {Raising bounded groups and splitting of radical extensions of commutative Banach algebras},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {85-98},
zbl = {0976.46038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv141i1p85bwm}
}
Bade, W.; Curtis, P.; Sinclair, A. Raising bounded groups and splitting of radical extensions of commutative Banach algebras. Studia Mathematica, Tome 141 (2000) pp. 85-98. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv141i1p85bwm/
[000] [AlEr] E. Albrecht and O. Ermert, Commutative nilpotent extensions of commutative C*-algebras split, Bull. London Math. Soc. 29 (1997), 601-608. | Zbl 0914.46044
[001] [BC1] W. G. Bade and P. C. Curtis, Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. | Zbl 0093.12503
[002] [BC2] W. G. Bade and P. C. Curtis, The Wedderburn decomposition of commutative Banach algebras, ibid., 851-866. | Zbl 0099.31903
[003] [BDL] W. G. Bade, H. G. Dales and Z. A. Lykova, Algebraic and strong splittings of extensions of Banach algebras, Mem. Amer. Math. Soc. 656 (1999). | Zbl 0931.46034
[004] [BD1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York, 1973. | Zbl 0271.46039
[005] [BD2] F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge Univ. Press, 1971. | Zbl 0207.44802
[006] [BD3] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Ser. 10, Cambridge Univ. Press, 1973.
[007] [Cu1] P. C. Curtis, Jr., Complementation problems concerning the radical of a commutative amenable Banach algebra, in: Proc. Centre Math. Anal. Austral. Nat. Univ. 21, 1989, 56-60.
[008] [Cu2] P. C. Curtis, Amenability, weak amenability and the close homomorphism property for commutative Banach algebras, in: Function Spaces: The Second Conference, Lecture Notes in Pure and Appl. Math. 172, Marcel Dekker, New York, 1995, 59-69. | Zbl 0846.46032
[009] [Cu1] P. C. Curtis, Jr., and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), 89-104.
[010] [Da] H. G. Dales, A discontinuous homomorphism from C(X), Amer. J. Math. 101 (1979), 647-734. | Zbl 0417.46054
[011] [Di] P. Dixon, Topologically nilpotent Banach algebras and factorization, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), 329-341. | Zbl 0762.46039
[012] [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.
[013] [GL] E. A. Gorin and V. Ya. Lin, On a condition on the radical of a Banach algebra ensuring strong decomposability, Mat. Zametki 2 (1967), 589-592 (in Russian).
[014] [Gr] F. P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand Math. Stud. 16, Van Nostrand, New York, 1969.
[015] [He] A. Ya. Helemskiĭ, Flat Banach modules and amenable algebras, Trudy Moskov. Mat. Obshch. 47 (1984), 179-218 (in Russian); English transl.: Amer. Math. Soc. Transl. 124, 1984, 199-224.
[016] [HP] E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.
[017] [Ho] G. Hochschild, On the cohomology theory for associative algebras, Ann. of Math. 46 (1945), 58-76.
[018] [Jo] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
[019] [Ka] H. Kamowitz, Cohomology groups of commutative Banach algebras, Trans. Amer. Math. Soc. 102 (1962), 352-372. | Zbl 0114.31703
[020] [Lu] G. Lumer, Spectral operators, Hermitian operators and bounded groups, Acta Sci. Math. (Szeged) 25 (1964), 75-85. | Zbl 0168.12103
[021] [P] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Volume I: Algebras and Banach Algebras, Encyclopedia Math. Appl. 49, Cambridge Univ. Press, Cambridge, New York, 1994. | Zbl 0809.46052
[022] [Sa] S. Saeki, Helson sets which disobey spectral synthesis, Proc. Amer. Math. Soc. 47 (1975), 371-377. | Zbl 0297.43007
[023] [So1] M. Solovej, Wedderburn decompositions and cohomology for Banach algebras, thesis, Univ. of Leeds, 1993.
[024] [So2] M. Solovej, Wedderburn decompositions of commutative Banach algebras, Proc. Amer. Math. Soc. 123 (1995), 3305-3315. | Zbl 0844.46027