Annals of Functional Analysis

A note on peripherally multiplicative maps on Banach algebras

Francois Schulz

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Banach algebras spectrum peripheral spectrum nonlinear preservers peripherally multiplicative maps


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.

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