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

Sen Huang

Studia Mathematica (1999)

  • Volume: 132, Issue: 1, page 37-69
  • ISSN: 0039-3223

Abstract

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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 ) ∩ , 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 .

How to cite

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Huang, Sen. "A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras." Studia Mathematica 132.1 (1999): 37-69. <http://eudml.org/doc/216585>.

@article{Huang1999,
abstract = {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 $Λ_T=σ(T)∩$, 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 .},
author = {Huang, Sen},
journal = {Studia Mathematica},
keywords = {automorphism; group representation; spectral analysis; continuous group representation; commutative Banach algebra; Gelfand space; continuous automorphisms; locally compact Abelian group; weak-star spectrum; unitary spectrum},
language = {eng},
number = {1},
pages = {37-69},
title = {A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras},
url = {http://eudml.org/doc/216585},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Huang, Sen
TI - A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 37
EP - 69
AB - 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 $Λ_T=σ(T)∩$, 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 .
LA - eng
KW - automorphism; group representation; spectral analysis; continuous group representation; commutative Banach algebra; Gelfand space; continuous automorphisms; locally compact Abelian group; weak-star spectrum; unitary spectrum
UR - http://eudml.org/doc/216585
ER -

References

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  1. [1] C. A. Akemann and P. A. Ostrand, The spectrum of a derivation of a*-algebra, J. London Math. Soc. 13 (1976), 525-530. Zbl0344.46119
  2. [2] W. Arveson, On group of automorphisms of operator algebras, J. Funct. Anal. 15 (1974), 217-243. 
  3. [3] A. Atzmon, On the existence of hyperinvariant subspaces, J. Operator Theory 11 (1984), 3-40. Zbl0583.47009
  4. [4] O. Bratteli, Derivations, Dissipations and Group Actions on C*-algebras, Springer, Berlin, 1986. 
  5. [5] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon & Breach, New York, 1968. Zbl0189.44201
  6. [6] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973), 133-252. Zbl0274.46050
  7. [7] A. Connes, Noncommutative Geometry, Academic Press, 1994. 
  8. [8] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math. 96 (1956), 1-66. Zbl0071.11302
  9. [9] I. Erdelyi and S.-W. Wang, A Local Spectral Theory for Closed Operators, Cambridge Univ. Press, Cambridge, 1985. Zbl0577.47035
  10. [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1979. Zbl0416.43001
  11. [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. [12] B. E. Johnson, Automorphisms of commutative Banach algebras, Proc. Amer. Math. Soc. 40 (1973), 497-499. Zbl0268.46047
  13. [13] H. Kamowitz and S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Funct. Anal. 4 (1969), 268-276. Zbl0182.17703
  14. [14] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968. 
  15. [15] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974. Zbl0285.46024
  16. [16] R. Larsen, Banach Algebras: An Introduction, Marcel Dekker, New York, 1973. Zbl0264.46042
  17. [17] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand, Toronto, 1953. Zbl0052.11701
  18. [18] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986. Zbl0585.47030
  19. [19] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London, 1979. Zbl0416.46043
  20. [20] H. Reiter, Classical Harmonic Analysis and Locally Compact Abelian Groups, Oxford Univ. Press, Oxford, 1968. Zbl0165.15601
  21. [21] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962. Zbl0107.09603
  22. [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. [23] S. Scheinberg, The spectrum of an automorphism, Bull. Amer. Math. Soc. 78 (1972), 621-623. Zbl0257.46057
  24. [24] E. Stοrmer, Spectra of ergodic transformations, J. Funct. Anal. 15 (1974), 202-215. Zbl0276.46031
  25. [25] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, Berlin, 1982. 

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