Continuous functional calculus for bounded self adjoint operators

The spectrum $\sigma (T)$ is a nonempty compact subset of $\mathbb{R}$.

The space $C(\sigma (T))$ with pointwise operations, conjugation and the supremum norm is a unital commutative C*-algebra: pointwise products and conjugation satisfy the algebra axioms, the supremum norm is submultiplicative and satisfies $\mathrm{\parallel }\bar{f}f{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}^2$, and completeness follows because a supremum-norm Cauchy sequence of continuous functions has pointwise limits by completeness of $\mathbb{C}$, converges uniformly by the standard estimate, and has continuous limit; $A$ is a unital, self-adjoint and point-separating function algebra inside it, hence uniformly dense.

The map $\mathrm{\Phi }:A\to \mathcal{B}(H)$, $p{\mathrm{\mid }}_{\sigma (T)}\mapsto p(T)$, is well defined and is complex-linear, multiplicative, unital, star-preserving (for $\sigma (T)\subseteq \mathbb{R}$ the conjugate of $p{\mathrm{\mid }}_{\sigma (T)}$ is $\bar{p}{\mathrm{\mid }}_{\sigma (T)}$) and isometric. For the star identity, write $p(z)={\sum }_{j=0}^m{a}_j{z}^j$: the C*-involution laws and $T^{\ast }=T$ give $p(T{)}^{\ast }={\sum }_{j=0}^m\overline{a_j}(T^{\ast }{)}^j={\sum }_{j=0}^m\overline{a_j}{T}^j$; on the real spectrum this polynomial equals $\overline{p(z)}$.

For $f\in C(\sigma (T))$ and polynomials $p_n$ with $\mathrm{\parallel }{p}_n-f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}(n+1)$ (which exist by density) the sequence $p_n(T)$ is Cauchy in operator norm because $\mathrm{\parallel }{p}_n(T)-p_m(T)\mathrm{\parallel }=\mathrm{\parallel }{p}_n-p_m{\mathrm{\parallel }}_{\mathrm{\infty }}\le \frac{1}{n+1}+\frac{1}{m+1}$, so it converges; the limit does not depend on the choice of sequence, since two such sequences differ in norm by at most $2\mathrm{/}(n+1)$.

Defining $\mathrm{\Psi }(f)$ as that limit makes $\mathrm{\Psi }:C(\sigma (T))\to \mathcal{B}(H)$ complex-linear, unital, multiplicative, star-preserving and isometric: each property holds for polynomial representatives by step 1.3 and passes to the limit by continuity of the algebra operations and the norm in $\mathcal{B}(H)$, while the norm identity passes by continuity of the modulus; moreover $\mathrm{\Psi }(z)=T$.

The range of $\mathrm{\Psi }$ is $C^{\ast }(I,T)$: each $\mathrm{\Psi }(f)$ is a norm limit of operators $p_n(T)$ lying in the unital $\ast$-algebra generated by $T$, so the range is contained in its closure; conversely $\mathrm{\Phi }(p{\mathrm{\mid }}_{\sigma (T)})=p(T)$ shows that every $\ast$-polynomial lies in the range, and the range is closed because $\mathrm{\Psi }$ is isometric on the complete space established in step 1.2: a convergent sequence of images has Cauchy preimages, whose limit maps to its image limit, so it contains the closure.

With step 1.2, the map $\mathrm{\Psi }$ is an isometric unital star-homomorphism of complex C*-algebras in the sense of the definition, and $\mathrm{\Psi }(z)=T$.

$\mathrm{\Psi }$ is the only such map: if $\mathrm{\Xi }$ is an isometric unital star-homomorphism with $\mathrm{\Xi }(z)=T$, then $\mathrm{\Xi }(p{\mathrm{\mid }}_{\sigma (T)})=p(T)=\mathrm{\Psi }(p{\mathrm{\mid }}_{\sigma (T)})$ for every polynomial $p$, by multiplicativity, unitality and star-preservation; for $f\in C(\sigma (T))$ and approximating polynomials $p_n$ with $\mathrm{\parallel }{p}_n-f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}(n+1)$, continuity of both isometric maps gives $\mathrm{\Xi }(f)=\lim \mathrm{\Xi }(p_n)=\lim p_n(T)=\mathrm{\Psi }(f)$.

The map $f\mapsto f(T):=\mathrm{\Psi }(f)$ is therefore the unique isometric unital star-homomorphism $C(\sigma (T))\to \mathcal{B}(H)$ with $z\mapsto T$ and range $C^{\ast }(I,T)$.