# A generalization of peripherally-multiplicative surjections between standard operator algebras

Open Mathematics (2009)

• Volume: 7, Issue: 3, page 479-486
• ISSN: 2391-5455

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## Abstract

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Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some bijective bounded linear operators B 1;B 2 of X* onto Y.

## How to cite

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Takeshi Miura, and Dai Honma. "A generalization of peripherally-multiplicative surjections between standard operator algebras." Open Mathematics 7.3 (2009): 479-486. <http://eudml.org/doc/269375>.

@article{TakeshiMiura2009,
abstract = {Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some bijective bounded linear operators B 1;B 2 of X* onto Y.},
author = {Takeshi Miura, Dai Honma},
journal = {Open Mathematics},
keywords = {Standard operator algebra; Peripheral spectrum; Peripherally-multiplicative operator; Spectrum-preserving map; standard operator algebra; peripheral spectrum; peripherally-multiplicative operator; spectrum-preserving map},
language = {eng},
number = {3},
pages = {479-486},
title = {A generalization of peripherally-multiplicative surjections between standard operator algebras},
url = {http://eudml.org/doc/269375},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Takeshi Miura
AU - Dai Honma
TI - A generalization of peripherally-multiplicative surjections between standard operator algebras
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 479
EP - 486
AB - Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some bijective bounded linear operators B 1;B 2 of X* onto Y.
LA - eng
KW - Standard operator algebra; Peripheral spectrum; Peripherally-multiplicative operator; Spectrum-preserving map; standard operator algebra; peripheral spectrum; peripherally-multiplicative operator; spectrum-preserving map
UR - http://eudml.org/doc/269375
ER -

## References

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1. [1] Aupetit B., Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc., 2000, 62, 917–924 http://dx.doi.org/10.1112/S0024610700001514 Zbl1070.46504
2. [2] Aupetit B., du T. Mouton H., Spectrum preserving linear mappings in Banach algebras, Studia Math., 1994, 109, 91–100 Zbl0829.46039
3. [3] Brešar M., Šemrl P., Mappings which preserve idempotents, local automorphisms, and local derivations, Canad J. Math., 1993, 45, 483–496 Zbl0796.15001
4. [4] Brešar M., Šemrl P., Linear maps preserving the spectral radius, J. Funct. Anal., 1996, 142, 360–368 http://dx.doi.org/10.1006/jfan.1996.0153 Zbl0873.47002
5. [5] Cui J.-L., Hou J.-C., Additive maps on standard operator algebras preserving parts of the spectrum, J. Math. Anal. Appl., 2003, 282, 266–278 http://dx.doi.org/10.1016/S0022-247X(03)00146-X Zbl1042.47027
6. [6] Hatori O., Hino K., Miura T., Oka H., Peripherally monomial-preserving maps between uniform algebras, Mediterr. J. Math., 2009, 6, 47–60 http://dx.doi.org/10.1007/s00009-009-0166-5 Zbl1192.46049
7. [7] Hatori O., Miura T., Oka H., An example of multiplicatively spectrum-preserving maps between non-isomorphic semi-simple commutative Banach algebras, Nihonkai Math. J., 2007, 18, 11–15 Zbl1143.46027
8. [8] 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
9. [9] 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
10. [10] Hatori O., Miura T., Takagi H., Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, preprint Zbl1113.46047
11. [11] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205 Zbl1141.46324
12. [12] Hou J.-C., Li C.-K., Wong N.-C., Jordan isomorphisms and maps preserving spectra of certain operator products, Studia Math., 2008, 184, 31–47 http://dx.doi.org/10.4064/sm184-1-2 Zbl1134.47028
13. [13] Jafarian A.A., Sourour A., Spectrum preserving linear maps, J. Funct. Anal., 1986, 66, 255–261 http://dx.doi.org/10.1016/0022-1236(86)90073-X
14. [14] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed lipschitz algebras, preprint Zbl1220.46033
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, 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, 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8 Zbl1134.46030
18. [18] Miura T., Honma D., Shindo R., Divisibly norm-preserving maps between commutative Banach algebras, preprint Zbl1232.46048
19. [19] 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
20. [20] Molnár L., Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Math., Springer, 2006, 1895
21. [21] 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
22. [22] 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
23. [23] Šemrl P., Two characterizations of automorphisms on B(X), Studia Math., 1993, 105, 143–149 Zbl0810.47001
24. [24] Tonev T., Luttman A., Algebra isomorphisms between standard operator algebras, Studia Math., 2009, 191, 163–170 http://dx.doi.org/10.4064/sm191-2-4 Zbl1179.47035

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