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Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$.

The collection of compact subsets of $\mathbb{C}$ has a natural topology, the one given by Hausdorff distance. Then a natural question is: what are the points of continuity of $\sigma$ ?

When $\mathcal{A}=B(\mathcal{H})$ for some Hilbert space $\mathcal{H}$, the problem is solved by Conway and his colleagues, but I don't know whether it is solved for general Banach algebras.

After thinking about this for quite a while, I thought a reversed way might also be interesting (although much harder it seems). That is, what is the initial topology on $\mathcal{A}$ with respect to $\sigma$ ?

I do not know where to start so any suggestion is welcome. Thanks!

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I’d not say the topology of Hausdorff distance seems very “natural” to me. The question is about alternative topology on $\mathcal A$ and I can’t answer, just another look: in a usual topology of a Banach space $σ$ is a continuous mapping to the non-Haudorff topology on the collection of compact subsets. The base of said topology if formed by open sets $U_E = \{ K\ |\ K\cap E = \emptyset\}$ where $E$ (is yet another) compact subset in ℂ. –  Incnis Mrsi Aug 14 at 12:57

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