Continuous functional calculus

Remarks

Existence comes from the commutative Gelfand–Naimark theorem applied to the commutative C*-algebra $C^*(a, 1)$ generated by $a$ and $1$, whose character space is homeomorphic to $\sigma(a)$. Uniqueness holds because the polynomials in $z$ and $\overline{z}$ are dense in $C(\sigma(a))$, by the Stone–Weierstrass theorem. The map $\Phi_a$ is isometric, its image is $C^*(a, 1)$, and $\sigma(f(a)) = f(\sigma(a))$.

Example: $A = M_n(\C)$ and $a = U D U^*$ with $U$ unitary and $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$. Here $\sigma(a) = \set{\lambda_1, \ldots, \lambda_n}$ and $f(a) = U \operatorname{diag}(f(\lambda_1), \ldots, f(\lambda_n)) U^*$. This map is linear, multiplicative, preserves adjoints, sends $1$ to $I$ and $\iota$ to $a$.

Non-example: $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ in $M_2(\C)$ is not normal. Its spectrum is $\set{0}$, so $\iota$ is the zero function on $\sigma(a)$, and any linear map sends it to $0 \ne a$. So no such $\Phi_a$ exists, and normality cannot be dropped.