Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Maliheh Hosseini; Fereshteh Sady

Open Mathematics (2010)

  • Volume: 8, Issue: 5, page 878-889
  • ISSN: 2391-5455

Abstract

top
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ 0 and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ 1, −1 and a clopen subset K of c(B) such that for each f ∈ A, . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.

How to cite

top

Maliheh Hosseini, and Fereshteh Sady. "Multiplicatively and non-symmetric multiplicatively norm-preserving maps." Open Mathematics 8.5 (2010): 878-889. <http://eudml.org/doc/269046>.

@article{MalihehHosseini2010,
abstract = {Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ 0 and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ 1, −1 and a clopen subset K of c(B) such that for each f ∈ A, \[ Tf\left( y \right) = \left\lbrace \{\begin\{array\}\{c\}\eta \left( y \right)f\left( \{\phi \left( y \right)\} \right) y \in K, \hfill \\ - \frac\{\alpha \}\{\{\left| \alpha \right|\}\}\eta \left( y \right)\overline\{f\left( \{\phi \left( y \right)\} \right)\} y \in c\left( B \right)\backslash K \hfill \\ \end\{array\}\} \right. \] . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.},
author = {Maliheh Hosseini, Fereshteh Sady},
journal = {Open Mathematics},
keywords = {Banach function algebra; Range-preserving map; Multiplicatively norm-preserving; Peaking function; Peripheral range; Peripheral spectrum; range-preserving map; multiplicatively norm preserving; peaking function; peripheral range; peripheral spectrum},
language = {eng},
number = {5},
pages = {878-889},
title = {Multiplicatively and non-symmetric multiplicatively norm-preserving maps},
url = {http://eudml.org/doc/269046},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Maliheh Hosseini
AU - Fereshteh Sady
TI - Multiplicatively and non-symmetric multiplicatively norm-preserving maps
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 878
EP - 889
AB - Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ 0 and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ 1, −1 and a clopen subset K of c(B) such that for each f ∈ A, \[ Tf\left( y \right) = \left\lbrace {\begin{array}{c}\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha }{{\left| \alpha \right|}}\eta \left( y \right)\overline{f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{array}} \right. \] . In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.
LA - eng
KW - Banach function algebra; Range-preserving map; Multiplicatively norm-preserving; Peaking function; Peripheral range; Peripheral spectrum; range-preserving map; multiplicatively norm preserving; peaking function; peripheral range; peripheral spectrum
UR - http://eudml.org/doc/269046
ER -

References

top
  1. [1] Araujo J., Font J.J., On Šilov boundaries for subspaces of continuous functions, Topology Appl., 1997, 77(2), 79–85 http://dx.doi.org/10.1016/S0166-8641(96)00132-0 Zbl0870.54018
  2. [2] Browder A., Introduction to Function Algebras, Mathematics Lecture Note Series, W.A. Benjamin, New York-Amsterdam, 1969 Zbl0199.46103
  3. [3] Burgos M., Jiménez-Vargas A., Villegas-Vallecillos M., Nonlinear conditions for weighted composition operators between Lipschitz algebras, J. Math. Anal. Appl., 2009, 359(1), 1–14 http://dx.doi.org/10.1016/j.jmaa.2009.05.017 Zbl1181.47037
  4. [4] Dales H.G., Boundaries and peak points for Banach function algebras, Proc. Lond. Math. Soc., 1971, 22, 121–136 http://dx.doi.org/10.1112/plms/s3-22.1.121 Zbl0211.15902
  5. [5] Hatori O., Miura T., Oka H., Takagi H., Peripheral multiplicativity of maps on uniformly closed algebras of continuous functions which vanish at infinity, Tokyo J. Math, 2009, 32(1), 91–104 http://dx.doi.org/10.3836/tjm/1249648411 Zbl1201.46046
  6. [6] Hatori O., Miura T., Takagi H., Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties, Proc. Amer. Math. Soc., 2006, 134(10), 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5 Zbl1102.46032
  7. [7] Hatori O., Miura T., Takagi H., Unital and multiplicatively spectrum-preserving surjections between 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 Zbl1113.46047
  8. [8] Hatori O., Miura T., Takagi H., Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, preprint available at http://arxiv.org/abs/0904.1939 
  9. [9] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205 Zbl1141.46324
  10. [10] Honma D., Norm-preserving surjections on algebras of continuous functions, Rocky Mountain J. Math., 2009, 39(5), 1517–1531 http://dx.doi.org/10.1216/RMJ-2009-39-5-1517 Zbl1183.46051
  11. [11] Hosseini M., Sady F., Multiplicatively range-preserving maps between Banach function algebras, J. Math. Anal. Appl., 2009, 357(1), 314–322 http://dx.doi.org/10.1016/j.jmaa.2009.04.008 Zbl1171.46021
  12. [12] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., (in press) Zbl1220.46033
  13. [13] 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 Zbl1178.46049
  14. [14] Lambert S., Spectral Preserver Problems in Uniform Algebras, Ph.D. thesis, University of Montana, Missoula, 2008 
  15. [15] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative mappings between uniform algebras, Contemp. Math., 2007, 435, 265–281 Zbl1148.46030
  16. [16] 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
  17. [17] 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 Zbl1134.46030
  18. [18] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2002, 130(1), 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X Zbl0983.47024
  19. [19] 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 Zbl1068.46028
  20. [20] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 2005, 48(1), 219–229 http://dx.doi.org/10.1017/S0013091504000719 Zbl1074.46033
  21. [21] Rao N.V., Tonev T.V., Toneva E.T., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416 Zbl1123.46035
  22. [22] Shindo R., Norm conditions for real-algebra isomorphisms between uniform algebras, Cent. Eur. J. Math., 2010, 8(1), 135–147 http://dx.doi.org/10.2478/s11533-009-0060-1 Zbl1201.47039
  23. [23] Stout E.L., The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson, 1971 Zbl0286.46049
  24. [24] Tonev T., Weakly multiplicative operators on function algebras without units, Banach Center Publ., (in press) 

NotesEmbed ?

top

You must be logged in to post comments.