A note on peripherally multiplicative maps on Banach algebras

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,y∈A, then we prove that the maps x↦ϕ1(1)ϕ2(x) and x↦ϕ1(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 x↦ϕ1(1)ϕ2(x) and x↦ϕ1(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.