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Kaplansky’s problem on invertibility preservers implying Jordan homomorphisms

Determine explicit conditions on unital complex Banach algebras A and B under which every linear unital invertibility preserver φ: A → B (possibly with additional assumptions such as surjectivity) is necessarily a Jordan homomorphism. This remains unresolved in general, including for the special case when A and B are C*-algebras.

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Background

The paper surveys classical and modern results on Jordan homomorphisms and linear preserver problems, highlighting the importance of characterizing when linear maps preserving certain properties must be Jordan homomorphisms. A central historical question in this direction was posed by Kaplansky in 1970.

The authors note that despite substantial progress in special cases, the general problem of identifying conditions on Banach algebras ensuring that unital linear invertibility preservers are Jordan homomorphisms has not been resolved, even within the widely studied class of C*-algebras.

References

These efforts trace back to at least 1970 and Kaplansky’s notable problem [21] which asks for conditions on unital (complex) Banach algebras A and B such that a linear unital invertibility preserver φ : A → B (possibly with some extra conditions, such as surjectivity) is necessarily a Jordan homomorphism. While significant attention has been given to this problem and progress has been made in specific cases, it remains largely unsolved, even for C -algebras (see e.g. [8, p. 270]).

— Jordan embeddings and linear rank preservers of structural matrix algebras (2409.16906 - Gogić et al., 25 Sep 2024) in Section 1.2 (Jordan homomorphisms)