Annals of Functional Analysis

A note on peripherally multiplicative maps on Banach algebras

Francois Schulz

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Abstract

Let A and B be complex Banach algebras, and let ϕ,ϕ1, and ϕ2 be surjective maps from A onto B. Denote by σ(x) the boundary of the spectrum of x. If A is semisimple, B has an essential socle, and σ(xy)=σ(ϕ1(x)ϕ2(y)) for each x,yA, then we prove that the maps xϕ1(1)ϕ2(x) and xϕ1(x)ϕ2(1) coincide and are continuous Jordan isomorphisms. Moreover, if A is prime with nonzero socle and ϕ1 and ϕ2 satisfy the aforementioned condition, then we show once again that the maps xϕ1(1)ϕ2(x) and xϕ1(x)ϕ2(1) coincide and are continuous. However, in this case we conclude that the maps are either isomorphisms or anti-isomorphisms. Finally, if A is prime with nonzero socle and ϕ is a peripherally multiplicative map, then we prove that ϕ is continuous and either ϕ or ϕ is an isomorphism or an anti-isomorphism.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 218-228.

Dates
Received: 22 April 2018
Accepted: 10 September 2018
First available in Project Euclid: 19 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1552960866

Digital Object Identifier
doi:10.1215/20088752-2018-0025

Mathematical Reviews number (MathSciNet)
MR3941383

Zentralblatt MATH identifier
07083890

Subjects
Primary: 47A10: Spectrum, resolvent
Secondary: 47B48: Operators on Banach algebras 47B49: Transformers, preservers (operators on spaces of operators)

Keywords
Banach algebras spectrum peripheral spectrum nonlinear preservers peripherally multiplicative maps

Citation

Schulz, Francois. A note on peripherally multiplicative maps on Banach algebras. Ann. Funct. Anal. 10 (2019), no. 2, 218--228. doi:10.1215/20088752-2018-0025. https://projecteuclid.org/euclid.afa/1552960866


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References

  • [1] B. Aupetit, A Primer On Spectral Theory, Universitext, Springer, New York, 1991.
  • [2] B. Aupetit and H. du T. Mouton, Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1994), no. 1, 91–100.
  • [3] B. Aupetit and H. du T. Mouton, Trace and determinant in Banach algebras, Studia Math. 121 (1996), no. 2, 115–136.
  • [4] A. Bourhim, J. Mashreghi, and A. Stepanyan, Maps between Banach algebras preserving the spectrum, Arch. Math. (Basel) 107 (2016), no. 6, 609–621.
  • [5] G. Braatvedt, R. Brits, and H. Raubenheimer, Spectral characterizations of scalars in a Banach algebra, Bull. Lond. Math. Soc. 41 (2009), no. 6, 1095–1104.
  • [6] D. D. Drăghia, Semi-simplicity of some semi-prime Banach algebras, Extracta Math. 10 (1995), no. 2, 189–193.
  • [7] I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, 1969.
  • [8] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261.
  • [9] I. Kaplansky, Algebraic and Analytic Aspects of Operator Algebras, CBMS-NSF Regional Conf. Ser. in Appl. Math. 1, Amer. Math. Soc., Providence, 1970.
  • [10] A. Luttman and T. Tonev, Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3589–3598.
  • [11] T. Miura and D. Honma, A generalization of peripherally-multiplicative surjections between standard operator algebras, Cent. Eur. J. Math. 7 (2009), no. 3, 479–486.
  • [12] L. Molnár, Some characterizations of the automorphisms of $B(H)$ and $C(X)$, Proc. Amer. Math. Soc. 130 (2002), no. 1, 111–120.
  • [13] F. Schulz and R. Brits, Uniqueness under spectral variation in the socle of a Banach algebra, J. Math. Anal. Appl. 444 (2016), no. 2, 1626–1639.
  • [14] F. Schulz, R. Brits, and G. Braatvedt, Trace characterizations and socle identifications in Banach algebras, Linear Algebra Appl. 472 (2015), 151–166.
  • [15] A. R. Sourour, Invertibility preserving linear maps on $\mathcal{L}(X)$, Trans. Amer. Math. Soc. 348 (1996), no. 1, 13–30.