An extension of the Gleason–Kahane–Żelazko theorem: A possible approach to Kaplanskyʹs problem

Let and be unital Banach algebras with semisimple. Is every surjective unital linear invertibility preserving map a Jordan homomorphism? This is a famous open question, often called “Kaplanskyʹs problem” in the literature. The Gleason–Kahane–Żelazko theorem gives an affirmative answer in the special case when . We obtain an improvement of this theorem. Our result implies that in order to answer the question in the affirmative it is enough to show that φ(x2) and φ(x) commute for every . In this way we obtain a new proof of the Marcus–Purves theorem.