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As is well known, the space $\mathcal{M}(\mathbb{T})$ of Borel complex measures on the torus $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$ is a (unital) commutative Banach algebra (with total variation as norm and convolution as multiplication). I've been trying to identify the maximal ideal space (that is the complex homomorphisms) of this algebra but so far couldn't even think of a reasonable candidate.

Using the Riesz representation theorem one gets that $\mathcal{M}(\mathbb{T})$ is isommetrical to $C(\mathbb{T})^*$ (the dual of the space of continuous functions from $\mathbb{T}$ to $\mathbb{C}$), so I started trying to figure out the bidual of $C(\mathbb{T})$ expecting it would be a reflexive space. Of course it isn't, but I had forgotten this.

I'm stuck with this. Can someone help me identifying this maximal ideal space? Any reference or idea would be appreciated too.

Community
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2 Answers2

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This space is truly humongous and it does not have a reasonable down-to-earth description.

One way to illustrate this claim is a recent result of Ohrysko, Wojciechowski and Graham which asserts that this space is non-separable. This result is surprising as the Gelfand space of $M(\mathbb{T})$ is canonically homeomorphic to a subset of the unit ball of $C[0,1]^*$, which in turn, is separable in the weak* topology.

However, there are abstract characterisations available. For example, this one due to Taylor. (See this paper by Ylinen for more references.)

I would like to know whether there exists a non-eventually-constant, convergent sequence in this space.

Tomasz Kania
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  • Great answer!! I was starting to think that it should be the case. Thank u, really appreciated :) –  Aug 28 '16 at 22:21
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It is so complicated that most researchers give up on this problem a long time ago. For details see discussion at section 4.3.14 in Banach and Locally Convex Algebras by A. Ya. Helemskii.

Norbert
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