Norm conditions for uniform algebra isomorphisms

Aaron Luttman; Scott Lambert

Open Mathematics (2008)

  • Volume: 6, Issue: 2, page 272-280
  • ISSN: 2391-5455

Abstract

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In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying T ( f ) T ( g ) + λ = f g + λ ∀ f , g ∈ A , then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism T ˜ : A → B e ⊕ B ¯ ( 1 - e ) such that T ( f ) = κ T ˜ ( f ) e + γ T ˜ ( f ) ¯ ( 1 - e ) for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.

How to cite

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Aaron Luttman, and Scott Lambert. "Norm conditions for uniform algebra isomorphisms." Open Mathematics 6.2 (2008): 272-280. <http://eudml.org/doc/269051>.

@article{AaronLuttman2008,
abstract = {In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying \[ \left\Vert \{T(f)T(g) + \lambda \} \right\Vert = \left\Vert \{fg + \lambda \} \right\Vert \forall f,g \in A, \] then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism \[ \tilde\{T\}:\{\rm A\} \rightarrow Be \oplus \bar\{B\}(1 - e) \] such that \[ T(f) = \kappa \left( \{\tilde\{T\}(f)e + \gamma \overline\{\tilde\{T\}(f)\} (1 - e)\} \right) \] for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.},
author = {Aaron Luttman, Scott Lambert},
journal = {Open Mathematics},
keywords = {uniform algebras; peripheral spectrum; isometric algebra isomorphism},
language = {eng},
number = {2},
pages = {272-280},
title = {Norm conditions for uniform algebra isomorphisms},
url = {http://eudml.org/doc/269051},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Aaron Luttman
AU - Scott Lambert
TI - Norm conditions for uniform algebra isomorphisms
JO - Open Mathematics
PY - 2008
VL - 6
IS - 2
SP - 272
EP - 280
AB - In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying \[ \left\Vert {T(f)T(g) + \lambda } \right\Vert = \left\Vert {fg + \lambda } \right\Vert \forall f,g \in A, \] then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism \[ \tilde{T}:{\rm A} \rightarrow Be \oplus \bar{B}(1 - e) \] such that \[ T(f) = \kappa \left( {\tilde{T}(f)e + \gamma \overline{\tilde{T}(f)} (1 - e)} \right) \] for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.
LA - eng
KW - uniform algebras; peripheral spectrum; isometric algebra isomorphism
UR - http://eudml.org/doc/269051
ER -

References

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  1. [1] Browder A., Introduction to function algebras, W.A. Benjamin Inc., New York-Amsterdam, 1969 Zbl0199.46103
  2. [2] Hatori O., Miura T., Takagi H., Characterization of isometric isomorphisms between uniform algebras via non-linear range preserving properties, Proc. Amer. Math. Soc., 2006, 134, 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5 Zbl1102.46032
  3. [3] Hatori O., Miura T., Takagi H., Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 2007, 326, 281–296 http://dx.doi.org/10.1016/j.jmaa.2006.02.084 Zbl1113.46047
  4. [4] Hatori O., Miura T., Takagi H., Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, preprint Zbl1113.46047
  5. [5] Hatori O., Miura T., Takagi H., Polynomially spectrum-preserving maps between commutative Banach algebras, preprint Zbl1113.46047
  6. [6] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205 Zbl1141.46324
  7. [7] Honma D., Norm-preserving surjections on algbras of continuous functions, Rocky Mountain J. Math., to appear 
  8. [8] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, preprint Zbl1220.46033
  9. [9] Kowalski S., Słodkowski Z., A Characterization of maximal ideals in commutative Banach algebras, Studia Math., 1980, 67, 215–223 Zbl0456.46041
  10. [10] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative operators between uniform algebras, Contemp. Math., 2007, 435, 265–281 Zbl1148.46030
  11. [11] Li C-K., Tsing N-K., Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl., 1992, 162/164, 217–235 http://dx.doi.org/10.1016/0024-3795(92)90377-M 
  12. [12] Luttman A., Tonev T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 2007, 135, 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8 Zbl1134.46030
  13. [13] Mazur S., Ulam S., Sur les transformations isométriques d’espaces vectoriels normés, C. R. Math. Acad. Sci. Paris, 1932, 194, 946–948 Zbl58.0423.01
  14. [14] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2001, 130, 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X Zbl0983.47024
  15. [15] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 2005, 133, 1135–1142 http://dx.doi.org/10.1090/S0002-9939-04-07615-4 Zbl1068.46028
  16. [16] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 2005, 48, 219–229 http://dx.doi.org/10.1017/S0013091504000719 Zbl1074.46033
  17. [17] Rao N.V., Tonev T., Toneva E., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416 Zbl1123.46035

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