Banach algebra mappings preserving the invertibility of differences

Let A and B be complex unital Banach algebras, and let φ,ψ:A→B be surjective mappings, with φ(0)=ψ(0)=0, which together preserve the invertibility of differences in both directions; that is, for any x,y∈A, x−y is invertible in A if and only if φ(x)−ψ(y) is invertible in B. If A is semisimple, we show that both φ and ψ preserve adjacency (differences of rank one) in both directions. By exploiting this connection we then prove, under the assumption that the socle of A is an infinite-dimensional, essential, and minimal two-sided ideal, that φ(x)=ψ(x)=uJ(x) for all x∈A, where u is a fixed invertible element in B and J:A→B is a linear or conjugate-linear Jordan-isomorphism.