 Character space of C1 on a compact interval

The coordinate function has spectrum $[0,1]$: outside the interval its difference from a scalar has a continuously differentiable reciprocal, and on the interval it has a zero. Every character of an algebra therefore takes the coordinate to some $t\in [0,1]$. Polynomial density in the norm containing both function and derivative, together with automatic continuity of characters, forces evaluation at $t$ on every function. The maximal ideals are exactly $\{f:f(t)=0\}$.