A note on peripherally multiplicative maps on Banach algebras

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 & map; ϕ₁(1)ϕ₂(x) and x ↦ ϕ₁(x)ϕ₂(1) coincide and are continuous Jordan isomorphisms. More- over, 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.