Spectral mapping for continuous normal functional calculus

The normal calculus $f\mapsto f(T)$ is an isometric unital star-isomorphism $C(\sigma (T))\to C^{\ast }(I,T)$ with $z\mapsto T$, so $f(T{)}^{\ast }=\bar{f}(T)$ and $f(T)g(T)=(fg)(T)$ (Continuous functional calculus for bounded normal operators).

The algebra $C^{\ast }(I,T)$ is commutative when $T$ is normal, and it is a unital C*-subalgebra of $\mathcal{B}(H)$ with the same identity (C star algebra generated by a normal operator as used by the calculus, C star algebra).

For a unital C*-subalgebra $B\subseteq A$ with the same identity one has ${\sigma }_B(b)={\sigma }_A(b)$ for every $b\in B$ (Spectral permanence for unital c star subalgebras).

For a nonzero commutative unital complex Banach algebra $C$ and $c\in C$ one has ${\sigma }_C(c)=\{\chi (c):\chi \in \mathrm{\Delta }(C)\}$ (Spectrum as character values).

For normal $T$ the map $\mathrm{\Phi }(\chi )=\chi (T)$ is a homeomorphism $\mathrm{\Delta }(C^{\ast }(I,T))\to \sigma (T)$. More precisely, $\chi \circ \mathrm{\Psi }$ is the evaluation character at the unique point $\mathrm{\Phi }(\chi )$, so $\chi (f(T))=f(\mathrm{\Phi }(\chi ))$ (Character space of generated normal algebra is operator spectrum, Characters of continuous functions are evaluations).

The operator spectrum of $f(T)\in C^{\ast }(I,T)\subseteq \mathcal{B}(H)$ is the spectrum in $\mathcal{B}(H)$ (Spectrum and resolvent of a bounded operator for the spectrum convention).

For a constant function $f\equiv c$, unitality and linearity give $f(T)=cI$; moreover $\lambda I-cI=(\lambda -c)I$ is boundedly invertible exactly when $\lambda \mathrm{\ne }c$, so $\sigma (cI)=\{c\}$ (Continuous functional calculus for bounded normal operators, Spectrum and resolvent of a bounded operator).

AC is the hypothesis of the permanence and character-space suppliers (The Axiom of Choice).