Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm.
In the 1980 paper The Gelfand space of the Banach algebra of Riemann integrable functions by Jörg Blatter (MR0602719) the author states, that no satisfactory representation is known for the space of maximal ideals of $R([0,1])$.
For the smaller space of regulated functions on $[0,1]$ the maximal ideal space consists of the functionals $\beta_x,\delta_x, \gamma_x$ given by $\beta_x(f)=f(x)$, $\delta_x(f)=\lim_{y\to x^+}f(y)$ and $\gamma_x(f)=\lim_{y\to x^-}f(y)$ where $x\in [0,1]$, see e.g. The character space of the algebra of regulated functions by S. K. Berberian (MR0487932).
I wonder whether some progress has been made concerning the maximal ideal space of $R([0,1])$ since Blatter's paper. In fact I would be very thankful if someone could provide me with an example of a character on $R([0,1])$ which is not of the form $\beta_x$, $\delta_x$, $\gamma_x$ as above or some intuition why the determination of the maximal ideal space of $R([0,1])$ is so much harder than the corresponding problem for the space of regulated functions.
