Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Antonio Jiménez-Vargas; Kristopher Lee; Aaron Luttman; Moisés Villegas-Vallecillos

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1197-1211
  • ISSN: 2391-5455

Abstract

top
Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all 𝕂 -valued Lipschitz functions on X - where 𝕂 is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = f(x): |f(x)| = ‖f‖∞ of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that R a n π ( T 1 ( f ) T 2 ( g ) ) ∩ R a n π ( S 1 ( f ) S 2 ( g ) ) ≠ ∅ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → 𝕂 with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.

How to cite

top

Antonio Jiménez-Vargas, et al. "Generalized weak peripheral multiplicativity in algebras of Lipschitz functions." Open Mathematics 11.7 (2013): 1197-1211. <http://eudml.org/doc/269721>.

@article{AntonioJiménez2013,
abstract = {Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb \{K\}$-valued Lipschitz functions on X - where $\mathbb \{K\}$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = f(x): |f(x)| = ‖f‖∞ of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb \{K\}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.},
author = {Antonio Jiménez-Vargas, Kristopher Lee, Aaron Luttman, Moisés Villegas-Vallecillos},
journal = {Open Mathematics},
keywords = {Lipschitz algebra; Peripheral multiplicativity; Spectral preservers; peripheral multiplicativity; spectral preservers},
language = {eng},
number = {7},
pages = {1197-1211},
title = {Generalized weak peripheral multiplicativity in algebras of Lipschitz functions},
url = {http://eudml.org/doc/269721},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Antonio Jiménez-Vargas
AU - Kristopher Lee
AU - Aaron Luttman
AU - Moisés Villegas-Vallecillos
TI - Generalized weak peripheral multiplicativity in algebras of Lipschitz functions
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1197
EP - 1211
AB - Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb {K}$-valued Lipschitz functions on X - where $\mathbb {K}$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = f(x): |f(x)| = ‖f‖∞ of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb {K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.
LA - eng
KW - Lipschitz algebra; Peripheral multiplicativity; Spectral preservers; peripheral multiplicativity; spectral preservers
UR - http://eudml.org/doc/269721
ER -

References

top
  1. [1] Browder A., Introduction to Function Algebras, W.A. Benjamin, New York-Amsterdam, 1969 Zbl0199.46103
  2. [2] Hatori O., Lambert S., Luttman A., Miura T., Tonev T., Yates R., Spectral preservers in commutative Banach algebras, Edwardsville, May 18–22, 2010, In: Function Spaces in Modern Analysis, Contemp. Math., 547, American Mathematical Socitey, Providence, 2011, 103–123 http://dx.doi.org/10.1090/conm/547/10812[Crossref] Zbl1239.46036
  3. [3] Hatori O., Miura T., Shindo R., Takagi T., Generalizations of spectrally multiplicative surjections between uniform algebras, Rend. Circ. Mat. Palermo, 2010, 59(2), 161–183 http://dx.doi.org/10.1007/s12215-010-0013-3[Crossref] Zbl1209.46027
  4. [4] 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[Crossref] Zbl1102.46032
  5. [5] 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(1), 281–296 http://dx.doi.org/10.1016/j.jmaa.2006.02.084[WoS][Crossref] Zbl1113.46047
  6. [6] Hatori O., Miura T., Takagi H., Polynomially spectrum-preserving maps between commutative Banach algebras, preprint available at http://arxiv.org/abs/0904.2322 Zbl1113.46047
  7. [7] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, In: Function Spaces, Edwardsville, May 16–20, 2006, Contemp. Math., 435, American Mathematical Society, Providence, 2007, 199–205 Zbl1141.46324
  8. [8] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., 2010, 40(3), 1903–1922 http://dx.doi.org/10.1216/RMJ-2010-40-6-1903[Crossref][WoS] Zbl1220.46033
  9. [9] Jiménez-Vargas A., Villegas-Vallecillos M., Lipschitz algebras and peripherally-multiplicative maps, Acta Math. Sin. (Engl. Ser.), 2008, 24(8), 1233–1242 http://dx.doi.org/10.1007/s10114-008-7202-4[Crossref] Zbl1178.46049
  10. [10] Lee K., Luttman A., Generalizations of weakly peripherally multiplicative maps between uniform algebras, J. Math. Anal. Appl., 2011, 375(1), 108–117 http://dx.doi.org/10.1016/j.jmaa.2010.08.051[Crossref] 
  11. [11] Luttman A., Lambert S., Norm conditions and uniform algebra isomorphisms, Cent. Eur. J. Math., 2008, 6(2), 272–280 http://dx.doi.org/10.2478/s11533-008-0016-x[WoS][Crossref] Zbl1151.46036
  12. [12] Luttman A., Tonev T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 2007, 135(11), 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8[WoS][Crossref] Zbl1134.46030
  13. [13] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2001, 130(1), 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X[Crossref] Zbl0983.47024
  14. [14] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 2005, 133(4), 1135–1142 http://dx.doi.org/10.1090/S0002-9939-04-07615-4[Crossref] Zbl1068.46028
  15. [15] Shindo R., Weakly-peripherally multiplicative conditions and isomorphisms between uniform algebras, Publ. Math. Debrecen, 2011, 78(3-4), 675–685 http://dx.doi.org/10.5486/PMD.2011.4937[Crossref][WoS] Zbl1274.46104
  16. [16] Tonev T., Yates R., Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl., 2009, 357(1), 45–53 http://dx.doi.org/10.1016/j.jmaa.2009.03.039[Crossref] Zbl1171.47032
  17. [17] Weaver N., Lipschitz Algebras, World Scientific, River Edge, 1999 http://dx.doi.org/10.1142/4100[Crossref] Zbl0936.46002

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.