I previously asked the following question of MathOverflow: [Showing that $\phi$ is a Jordan morphism](https://mathoverflow.net/questions/262496/showing-that-phi-is-a-jordan-morphism)
in which I was asking assistance with proving the following statement made in the introduction *Spectrum preservig linear mappings in Banach algerbas* by *B. Aupetit and H. du T. Mouton* (1994).



> 
> If $\phi$ is a linear mapping from a Banach algebra $A$ into another one $B$ such that $\phi(1)=1$ and $\phi(x)^{-1}=\phi(x^{-1})$ for $x$ invertible, then using exponentials it is easy to prove that $\phi$ is a Jordan morphism.
> 
> 
> 


However, while reading the paper *Spectral Characterization of Idempotents and Invertibility Preserving Linear Maps* by *M. Brešar* and *P. Šemrl* (1999), I came across the following:



> 
> Let $A$ and $B$ be semi-simple Banach algebras and $\phi:A \to B$ a unital bijective linear map preserving invertibility. It is then conjectured that $\phi$ must be a Jordan isomorphism.
> 
> 
> 


**My question:** When comparing the statements of the two papers, is the condition that $A$ and $B$ are semi-simple necessary to ensure that $\phi$ is Jordan?


I understand that the bijective condition of the latter statement, simply changes the (homo)morphism of the first statement into a (iso)morphism. I am simply interested in seeing how the condition that $A$ and $B$ being semi-simple comes into play.



Do we require $A$ and $B$ to be semi-simple?
I don't know the sharpest examples to answer your question, but the basic "idea" is that the conjecture you mention (due I think to Kaplansky?) is only reasonable when the Jacobson radical is trivial, because it is the part which spectral theory cannot detect.


To give a more concrete illustration: suppose $A$ is a unital CBA whose Jacobson radical $J$ has codimension $1$. (Take your favourite radical CBA and adjoin an identity). Now every element of the coset $1+J$ is invertible. So if you take any linear bijection $\theta:J\to J$ and extend it to a linear bijection $\phi:A\to A$ in the obvious way, then $\phi$ will be invertibility-preserving; provided $\dim(J)\geq 2$ it should be possible to cook up $\theta$ so that $\phi$ is not a homomorphism.


