Three spectral notions for representations of commutative Banach algebras

Yngve Domar; Lars-Ake Lindahl

Annales de l'institut Fourier (1975)

  • Volume: 25, Issue: 2, page 1-32
  • ISSN: 0373-0956

Abstract

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Let T be a bounded representation of a commutative Banach algebra B . The following spectral sets are studied. Λ 1 ( T ) : the Gelfand space of the quotient algebra B / Ker T . Λ 2 ( T ) : the Gelfand space of the operator algebra Im T . Λ 3 ( T ) : those characters φ of B for which the inequalities ∥ T b x - b ^ ( φ ) x ∥ < ϵ ∥ x ∥ , b ∈ F , have a common solution x ≠ 0 , for any ϵ > 0 and any finite subset F of B . A theorem of Beurling on the spectrum of L ∞ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for T the regular representation and its adjoint.

How to cite

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Domar, Yngve, and Lindahl, Lars-Ake. "Three spectral notions for representations of commutative Banach algebras." Annales de l'institut Fourier 25.2 (1975): 1-32. <http://eudml.org/doc/74222>.

@article{Domar1975,
abstract = {Let $T$ be a bounded representation of a commutative Banach algebra $B$. The following spectral sets are studied. $\Lambda _1(T)$: the Gelfand space of the quotient algebra $B/\{\rm Ker\}\ T$. $\Lambda _2(T)$: the Gelfand space of the operator algebra $\{\rm Im\}\ T$. $\Lambda _3(T)$: those characters $\varphi $ of $B$ for which the inequalities $\Vert T_bx - \hat\{b\}(\varphi )x\Vert &lt; \varepsilon \Vert x\Vert $, $b\in F$, have a common solution $x\ne 0$, for any $\varepsilon &gt;0$ and any finite subset $F$ of $B$. A theorem of Beurling on the spectrum of $L^\infty $-functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for $T$ the regular representation and its adjoint.},
author = {Domar, Yngve, Lindahl, Lars-Ake},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {1-32},
publisher = {Association des Annales de l'Institut Fourier},
title = {Three spectral notions for representations of commutative Banach algebras},
url = {http://eudml.org/doc/74222},
volume = {25},
year = {1975},
}

TY - JOUR
AU - Domar, Yngve
AU - Lindahl, Lars-Ake
TI - Three spectral notions for representations of commutative Banach algebras
JO - Annales de l'institut Fourier
PY - 1975
PB - Association des Annales de l'Institut Fourier
VL - 25
IS - 2
SP - 1
EP - 32
AB - Let $T$ be a bounded representation of a commutative Banach algebra $B$. The following spectral sets are studied. $\Lambda _1(T)$: the Gelfand space of the quotient algebra $B/{\rm Ker}\ T$. $\Lambda _2(T)$: the Gelfand space of the operator algebra ${\rm Im}\ T$. $\Lambda _3(T)$: those characters $\varphi $ of $B$ for which the inequalities $\Vert T_bx - \hat{b}(\varphi )x\Vert &lt; \varepsilon \Vert x\Vert $, $b\in F$, have a common solution $x\ne 0$, for any $\varepsilon &gt;0$ and any finite subset $F$ of $B$. A theorem of Beurling on the spectrum of $L^\infty $-functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for $T$ the regular representation and its adjoint.
LA - eng
UR - http://eudml.org/doc/74222
ER -

References

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  1. [1] R. ARENS, Extensions of Banach algebras, Pacific J. Math., 10 (1960), 1-16. Zbl0095.09801MR23 #A4031
  2. [2] A. BEURLING, Un théorème sur les fonctions bornées et uniformément continues sur l'axe réel, Acta Math., 77 (1945), 127-136. Zbl0061.13311MR7,61f
  3. [3] M.D. CHOI and C. DAVIS, The spectral mapping theorem for joint approximate point spectrum, Bull. Amer. Math. Soc., 80 (1974), 317-321. Zbl0276.47001MR48 #12104
  4. [4] J. DIXMIER, Sur un théorème de Banach, Duke Math. J., 15 (1948), 1057-1071. Zbl0031.36301MR10,306g
  5. [5] Y. DOMAR, Harmonic analysis based on certain commutative Banach algebras, Acta Math., 96 (1956), 1-66. Zbl0071.11302MR17,1228a
  6. [6] Y. DOMAR, On spectral analysis in the narrow topology, Math. Scand., 4 (1956), 328-332. Zbl0078.29401MR19,413b
  7. [7] Y. DOMAR, Some results on narrow spectral analysis, Math. Scand., 20 (1967), 5-18. Zbl0166.11201MR36 #690
  8. [8] V.P. GURARII, Spectral synthesis of bounded functions on the half-axis, Funkcional. Anal. i Prilozen, 3 n° 4 (1969), 34-48 (= Functional Anal. Appl., 3 (1969), 282-294). Zbl0205.42304MR41 #743b
  9. [9] L-Å. LINDAHL, On narrow spectral analysis, Math. Scand., 26 (1970), 149-164. Zbl0189.44601MR41 #5893
  10. [10] L-Å. LINDAHL, On ideals of joint topological divisors of zero, to appear in Studia Math., 53. Zbl0268.46049MR51 #13693
  11. [11] Yu. I. LYUBICH, On the spectrum of a representation of an abelian topological group, Dokl. Akad. Nauk. SSSR, 200 (1971), 777-780 (= Soviet Math. Dokl., 12 (1971), 1482-1486). Zbl0235.22008
  12. [12] Yu. I. LYUBICH, V.I. MATSAEV and G.M. FEL'DMAN, On representations with a separable spectrum, Funkcional. Anal. i Prilozen, 7 no. 2 (1973), 52-61 (= Functional Anal. Appl., 7 (1973), 129-136. Zbl0285.22004
  13. [13] B. NYMAN, On the one-dimensional translation group and semi-group in certain function spaces, Uppsala, (1950), 55 pp. (Thesis). Zbl0037.35401MR12,108g
  14. [14] C. RICKART, General theory of Banach algebras, Van Nostrand, (1960). Zbl0095.09702MR22 #5903
  15. [15] W. RUDIN, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc., 7 (1956), 808-811. Zbl0073.29701MR18,472c
  16. [16] Z. SLODKOWSKI, On ideals consisting of joint topological divisors of zero, Studia Math., 48 (1973), 83-88. Zbl0271.46046MR50 #1003
  17. [17] N. VAROPOULOS, Groups of continuous functions in harmonic analysis, Acta Math., 125 (1970), 109-154. Zbl0214.38102MR43 #7868
  18. [18] C.R. WARNER, Weak-* dense subspaces of L∞ (R), Math. Ann., 197 (1972), 180-181. Zbl0221.46027MR47 #2336
  19. [19] W. ŻELAZKO, On a certain class of non-removable ideals in Banach algebras, Studia Math., 44 (1972), 87-92. Zbl0213.40603MR47 #2376

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