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

TL;DR

This paper investigates the topological and cohomological properties of the Gelfand space of the measure algebra on the circle group, revealing non-separability and embedding of multiple copies of the Stone-Čech compactification.

Contribution

It provides new insights into the structure of the Gelfand space of the measure algebra on the circle, including non-separability and a novel method to embed copies of βℤ.

Findings

The Gelfand space of M(𝕋) is not separable.

Many copies of βℤ can be embedded in the Gelfand space.

The embeddings are not accessible via the canonical Fourier-Stieltjes transform.

Abstract

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).