Question: Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital.

I couldn't find any examples of such algebras.

Thanks!

Answer: 
The [Volterra algebra](http://mathworld.wolfram.com/VolterraAlgebra.html) $V$ is an example of a commutative Banach algebra without maximal ideals (hence with empty character space). See also Definition 4.7.38 in 

> 
> H. G. Dales, *Banach algebras and automatic continuity*, London Math. Soc.
>  Monographs, Volume 24, Clarendon Press, Oxford, 2000.
> 
> 
> 

Okay, let me prove this claim. This relies on three facts:

The algebra $V$ has a bounded approximate identity, *e.g.* $(n\cdot \mathbf{1}\_{\big[0, \tfrac{1}{n}\big]})\_{n=1}^\infty$, hence by the [Cohen factorisation theorem](http://en.wikipedia.org/wiki/Cohen%E2%80%93Hewitt_factorization_theorem), $V = V^2$. Consequently, [all maximal ideals of $V$ (if exist) are closed](https://math.stackexchange.com/questions/859247/maximal-ideals-and-maximal-subspaces-of-normed-algebras/863704#863704). 

Now apply a result of Dixmier which tells you that no prime ideal of $V$ is closed. Of course, maximal ideals are prime so, the conclusion follows. You will find the proof of Dixmier's result in the above-mentioned book by Dales (Theorem 4.7.58).
A related comment: I think I have an example of a commutative unital complex algebra $A$ without 
any nontrivial complex homomorphisms. Take 
$A$ to be all rational functions with complex coefficients, that is 
$$
A= \{[p/q]: p,q \text{ complex polynomials and }q \text{ not identically } 0\}.
$$
Here $[p/q]$ denotes the equivalence class of $(p,q)$ under the relation
$(p,q)\sim(r,s)$ if $ps=qr$. Then $A$ is a commutative unital complex algebra (which is also a field, and the only maximal ideal is $0$). But there is no nontrivial complex homomorphism $\varphi:A \rightarrow \mathbb{C}$, because if $\varphi(z)=\alpha$, then 
$$
\varphi\Big(\frac{1}{z-\alpha}\Big)=\frac{1}{\varphi(z-\alpha)}=\frac{1}{0},
$$ 
a contradiction.