Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define to be the union of all sets where a ∈ A, and to be the closure of the union of all sets where φ ∈ ∆(A), and call the unitary spectrum of α. Starting by showing that the closure of (for fixed φ ∈ ∆(A)) is a subsemigroup of Ĝ we characterize the structure properties of the group representation α such as norm continuity, growth and existence of non-trivial invariant subspaces through its unitary spectrum For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by for all n ∈ ℤ. It is shown that , where σ(T) is the spectrum of T and is the unit circle. From this fact we give an easy proof of the Kamowitz-Scheinberg theorem which asserts that the spectrum σ(T) either contains or is a finite union of finite subgroups of .
@article{bwmeta1.element.bwnjournal-article-smv132i1p37bwm,
author = {Sen Huang},
title = {A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {37-69},
zbl = {0984.46029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv132i1p37bwm}
}
Huang, Sen. A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras. Studia Mathematica, Tome 133 (1999) pp. 37-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv132i1p37bwm/
[00000] [1] C. A. Akemann and P. A. Ostrand, The spectrum of a derivation of a*-algebra, J. London Math. Soc. 13 (1976), 525-530. | Zbl 0344.46119
[00001] [2] W. Arveson, On group of automorphisms of operator algebras, J. Funct. Anal. 15 (1974), 217-243.
[00002] [3] A. Atzmon, On the existence of hyperinvariant subspaces, J. Operator Theory 11 (1984), 3-40. | Zbl 0583.47009
[00003] [4] O. Bratteli, Derivations, Dissipations and Group Actions on C*-algebras, Springer, Berlin, 1986.
[00004] [5] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon & Breach, New York, 1968. | Zbl 0189.44201
[00005] [6] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973), 133-252. | Zbl 0274.46050
[00006] [7] A. Connes, Noncommutative Geometry, Academic Press, 1994.
[00007] [8] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math. 96 (1956), 1-66. | Zbl 0071.11302
[00008] [9] I. Erdelyi and S.-W. Wang, A Local Spectral Theory for Closed Operators, Cambridge Univ. Press, Cambridge, 1985. | Zbl 0577.47035
[00009] [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1979. | Zbl 0416.43001
[00010] [11] S.-Z. Huang, Spectral theory for non-quasianalytic representations of locally compact abelian groups, thesis, Universität Tübingen, 1996. A complete summary is given in "Dissertation Summaries in Mathematics" 1 (1996), 171-178.
[00011] [12] B. E. Johnson, Automorphisms of commutative Banach algebras, Proc. Amer. Math. Soc. 40 (1973), 497-499. | Zbl 0268.46047
[00012] [13] H. Kamowitz and S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Funct. Anal. 4 (1969), 268-276. | Zbl 0182.17703
[00013] [14] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.
[00014] [15] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974. | Zbl 0285.46024
[00015] [16] R. Larsen, Banach Algebras: An Introduction, Marcel Dekker, New York, 1973. | Zbl 0264.46042
[00016] [17] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand, Toronto, 1953. | Zbl 0052.11701
[00017] [18] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986. | Zbl 0585.47030
[00018] [19] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London, 1979. | Zbl 0416.46043
[00019] [20] H. Reiter, Classical Harmonic Analysis and Locally Compact Abelian Groups, Oxford Univ. Press, Oxford, 1968. | Zbl 0165.15601
[00020] [21] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962. | Zbl 0107.09603
[00021] [22] S. Scheinberg, Automorphisms of commutative Banach algebras, in: Problems in Analysis, R. C. Gunning (ed.), Princeton Univ. Press, Princeton, N.J., 1971, 319-323.
[00022] [23] S. Scheinberg, The spectrum of an automorphism, Bull. Amer. Math. Soc. 78 (1972), 621-623. | Zbl 0257.46057
[00023] [24] E. Stοrmer, Spectra of ergodic transformations, J. Funct. Anal. 15 (1974), 202-215. | Zbl 0276.46031
[00024] [25] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, Berlin, 1982.