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](https://projecteuclid.org/euclid.afm/1485802756), 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. 



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> Does $\Delta$ have any injective convergent sequences?
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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? 



Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra?
Norbert is right (see the paper *[The Structure of Convolution Measure Algebras)](http://The%20Structure%20of%20Convolution%20Measure%20Algebras)*. Actually, I had known the result but it didn't occur to me that it would have solved the problem. 


The maximal ideal space of $M(\mathbb T)$ contains an analytic disk, which in particular, is a homeomorphic copy of $\mathbb D$, so there are plenty of convergent sequences therein.


