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Maybe I would even go a bit further and say that the consequences of the Hahn-Banach theorem are not part of the percieved reality. Nobody has ever encountered an invariant mean on ${\mathbb Z}$! On the other side, nobody ever found it necessary to consider non-continuous linear operators from one Hilbert space (defined everywhere) to another. Doesn't it seem more realistic to strive for continuity of all such operators and not for the existence of the mean. Of course, it is difficult to come with a definition.
For most purposes, only separable Banach spaces are relevant. Cannot Hahn-Banach can be proved with Countable Choice of Dependent Choice in this case? Of course, non-separable Banach spaces such as $\ell^{\infty} {\mathbb N}$ are relevant. For example, I see that for the existence of Banach limits or means on amenable groups (both are elements in the dual of $\ell^{\infty} {\mathbb N}$), Krein-Milman (which uses Tychonoff) is practical and seems right. However, the non-constructive nature of the mean and the unintuitive consequences give me the feeling that these objects are not realistic.
It remains interesting to ask the question for other $C^*$-algebras or operator algebras, such as the algebra of upper triangular operators on $\ell^2 {\mathbb N}$.
Right, under some conditions on the element $x$, Hille can show that the exponential map is open at $x$. His conditions are sufficient but probably not necessary. If the answer to the Question is negative, one should ask about a precise characterization of the elements where $\exp$ is open.
I was thinking about the other direction of my claim about zero-divisors and nilpotent elements. In fact I was just explaining some detail of Passman's argument that you mentioned already in your comment.
