This expository article deals with results surrounding the following question: Which pairs of Banach algebras A and B have the property that every unital invertibility preserving linear map from A to B is a Jordan homomorphism?
Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.
Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there...
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