On the Gelfand space of the measure algebra on the circle group

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).