Gleason–Kahane–Żelazko Theorem for Bilinear Maps

Let A and B be two unital Banach algebras and 𝔘 = A × B. We prove that the bilinear mapping φ: 𝔘 → ℂ is a bi-Jordan homomorphism if and only if φ is unital, invertibility preserving and jointly continuous. Additionally, if A is commutative, then φ is a bi-homomorphism.