  This paper is devoted to studying certain topological properties of the
maximal ideal space of the measure algebra on the circle group. In particular,
we focus on Cech cohomologies of this space. Moreover, we show that the Gelfand
space of $M(\mathbb{T})$ is not separable. On the other hand, we give a direct
procedure to recover many copies of $\beta\mathbb{Z}$ in
$\mathfrak{M}(M(\mathbb{T}))$, but we also show that this is result is not
accessible in the most natural way (namely, by the canonical mapping induced by
homomorphism assigning to measure its Fourier - Stieltjes transform).
