# Norm conditions for uniform algebra isomorphisms

Open Mathematics (2008)

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

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topAaron 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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