Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 4 / d / Solution

Apply part (c) to the commutative unital C-star algebra $A=C^{\ast }(I,T)$, commutative because $T$ is normal. A character is determined by $z=\chi (T)$, since $T,T^{\ast }$ generate the algebra and $\chi (T^{\ast })=\overline{z}$. The spectrum identity in part (c) shows that every $z\in {\sigma }_A(T)$ occurs. Thus $X\to {\sigma }_A(T)$ is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. Spectral permanence identifies this spectrum with the operator spectrum.

For any complex polynomial $p$ in two commuting variables, evaluation gives $\chi (p(T,T^{\ast }))=p(z,\overline{z})$. Again using the spectrum identity and spectral permanence,This proves the requested spectral mapping theorem, including constants, and explains why conjugation in the second coordinate is required.