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## Studia Mathematica

1999 | 132 | 1 | 37-69
Tytuł artykułu

### A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $φ_a(t):=φ(α_t a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_w*(φ_a)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_φ^a$ to be the union of all sets $σ_w*(φ_a)$ where a ∈ A, and $Λ_α$ to be the closure of the union of all sets $Ʌ_φ^a$ where φ ∈ ∆(A), and call $Λ_α$ the unitary spectrum of α. Starting by showing that the closure of $Ʌ_φ^a$ (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 $Λ_α.$ For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by $T_n:=T^n$ for all n ∈ ℤ. It is shown that $Λ_T=σ(T)∩𝕋$, where σ(T) is the spectrum of T and 𝕋 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 𝕋 or is a finite union of finite subgroups of 𝕋.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
37-69
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-04-04
poprawiono
1998-03-02
Twórcy
autor
Bibliografia
• [1] C. A. Akemann and P. A. Ostrand, The spectrum of a derivation of a*-algebra, J. London Math. Soc. 13 (1976), 525-530.
• [2] W. Arveson, On group of automorphisms of operator algebras, J. Funct. Anal. 15 (1974), 217-243.
• [3] A. Atzmon, On the existence of hyperinvariant subspaces, J. Operator Theory 11 (1984), 3-40.
• [4] O. Bratteli, Derivations, Dissipations and Group Actions on C*-algebras, Springer, Berlin, 1986.
• [5] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon & Breach, New York, 1968.
• [6] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973), 133-252.
• [7] A. Connes, Noncommutative Geometry, Academic Press, 1994.
• [8] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math. 96 (1956), 1-66.
• [9] I. Erdelyi and S.-W. Wang, A Local Spectral Theory for Closed Operators, Cambridge Univ. Press, Cambridge, 1985.
• [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1979.
• [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.
• [12] B. E. Johnson, Automorphisms of commutative Banach algebras, Proc. Amer. Math. Soc. 40 (1973), 497-499.
• [13] H. Kamowitz and S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Funct. Anal. 4 (1969), 268-276.
• [14] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.
• [15] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974.
• [16] R. Larsen, Banach Algebras: An Introduction, Marcel Dekker, New York, 1973.
• [17] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand, Toronto, 1953.
• [18] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986.
• [19] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London, 1979.
• [20] H. Reiter, Classical Harmonic Analysis and Locally Compact Abelian Groups, Oxford Univ. Press, Oxford, 1968.
• [21] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962.
• [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.
• [23] S. Scheinberg, The spectrum of an automorphism, Bull. Amer. Math. Soc. 78 (1972), 621-623.
• [24] E. Stοrmer, Spectra of ergodic transformations, J. Funct. Anal. 15 (1974), 202-215.
• [25] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, Berlin, 1982.
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Bibliografia
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