# Norm conditions for real-algebra isomorphisms between uniform algebras

Open Mathematics (2010)

• Volume: 8, Issue: 1, page 135-147
• ISSN: 2391-5455

top

## Abstract

top
Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism $\stackrel{˜}{S}$ : A → B such that $\stackrel{˜}{S}$ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.

## How to cite

top

Rumi Shindo. "Norm conditions for real-algebra isomorphisms between uniform algebras." Open Mathematics 8.1 (2010): 135-147. <http://eudml.org/doc/269806>.

@article{RumiShindo2010,
abstract = {Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism $\tilde\{S\}$ : A → B such that $\tilde\{S\}$ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.},
author = {Rumi Shindo},
journal = {Open Mathematics},
keywords = {Banach algebra; Uniform algebra; Norm-preserving; uniform algebra; isomorphisms; norm preserving},
language = {eng},
number = {1},
pages = {135-147},
title = {Norm conditions for real-algebra isomorphisms between uniform algebras},
url = {http://eudml.org/doc/269806},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Rumi Shindo
TI - Norm conditions for real-algebra isomorphisms between uniform algebras
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 135
EP - 147
AB - Let A and B be uniform algebras. Suppose that α ≠ 0 and A 1 ⊂ A. Let ρ, τ: A 1 → A and S, T: A 1 → B be mappings. Suppose that ρ(A 1), τ(A 1) and S(A 1), T(A 1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A 1, S(e 1)−1 ∈ S(A 1) and S(e 1) ∈ T(A 1) for some e 1 ∈ A 1 with ρ(e 1) = 1, then there exists a real-algebra isomorphism $\tilde{S}$ : A → B such that $\tilde{S}$ (ρ(f)) = S(e 1)−1 S(f) for every f ∈ A 1. We also give some applications of this result.
LA - eng
KW - Banach algebra; Uniform algebra; Norm-preserving; uniform algebra; isomorphisms; norm preserving
UR - http://eudml.org/doc/269806
ER -

## References

top
1. [1] Browder A., Introduction to function algebras, W.A. Benjamin, 1969 Zbl0199.46103
2. [2] Hatori O., Miura T., Takagi H., Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving property, 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., Hino K., Miura T., Oka H., Peripherally monomial-preserving maps between uniform algebras, Mediterr. J. Math., 2009, 6, 47–59 http://dx.doi.org/10.1007/s00009-009-0166-5 Zbl1192.46049
6. [6] Hatori O., Miura T., Shindo R., Takagi H., Generalizations of spectrally multiplicative surjections between uniform algebras, preprint Zbl1209.46027
7. [7] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2006,435, 199–205
8. [8] Honma D., Norm-preserving surjections on algebras of continuous functions, Rocky Mountain J. Math., to appear Zbl1183.46051
9. [9] Kelley J.L., General topology, D. Van Nostrand Company (Canada), 1955
10. [10] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281 Zbl1148.46030
11. [11] 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
12. [12] Luttman A., Lambert S., Norm conditions for uniform algebra isomorphisms, Cent. Eur. J. Math., 2008, 6(2), 272–280 http://dx.doi.org/10.2478/s11533-008-0016-x Zbl1151.46036
13. [13] Miura T., Honma D., Shindo R., Divisibly norm-preserving maps between commutative Banach algebras, Rocky Mountain J. Math., to appear Zbl1232.46048
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. Edinburgh Math. Soc., 2005, 48,219–229 http://dx.doi.org/10.1017/S0013091504000719 Zbl1074.46033
17. [17] Shindo R., Maps between uniform algebras preserving norms of rational functions, preprint Zbl1222.46039

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.