Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal space of $M(\mathbb T)$. This space is quite large–recently it was shown that $\Delta$ is non-separable, for example. It also has many copies of $\beta \mathbb N$, yet it is not extremely disconnected, however, in certain sense, it is not too far from being so.
Extremely disconnected spaces do not have injective convergent sequences, again $\beta \mathbb N$ serves a paradigm example.
Does $\Delta$ have any injective convergent sequences?
I have some evidence that it should not be the case but I am unable to make it into a proof. Maybe this follows from some known facts anyway?