Continuous functional calculus ($f\mapsto f(a)$)

For a Normal element of a C-star algebra $a$ in a unital C-star algebra, the Commutative Gelfand--Naimark theorem identifies $C(\sigma (a))$ isometrically with the closed unital algebra generated by $a$ and $a^{\ast }$. The coordinate function maps to $a$, defining $f(a)$ for every continuous $f$ on the spectrum of an element.

For a normal C-star element $x$ in a nonunital algebra, take its spectrum of an element in the C-star unitization. Zero belongs to that spectrum. Continuous functions on this compact spectrum that vanish at zero form an ideal canonically identified with the displayed function algebra. Their values under the unital continuous functional calculus belong to the original algebra because the scalar quotient evaluates them at zero. Star-polynomial approximation with the constant term removed gives exactly the image $C^{\ast }(x)$.

For a Positive element of a C-star algebra $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\Vert x\Vert t^{-3/2}$. The scalar identity follows by setting $t=s{u}^2$ for $s>0$ and using ${\int }_0^{\infty }2/(1+u^2)du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact spectrum of an element, so continuous functional calculus gives the operator identity.

For Positive elements of a C-star algebra $x\le y$, inversion reverses the order of strictly positive elements, so $(t1+x{)}^{-1}\ge (t1+y{)}^{-1}$ for $t>0$. Subtracting their square-root resolvent integral in a C-star algebra formulas givesThe integrand is positive, and the positive cone is closed, so the norm limit of these integrals is positive. No commutativity of $x$ and $y$ is required. In particular, $b^2\ge a^2$ for positive $a,b$ implies $b\ge a$, even though squaring itself is not generally order preserving.

For bounded self-adjoint $a_i,a$, no uniform bound on $\Vert a_i\Vert$ is needed. The resolvent identity gives $(a_i-i{)}^{-1}-(a-i{)}^{-1}=(a_i-i{)}^{-1}(a-a_i)(a-i{)}^{-1}$, and the first factor has norm at most one. Thus both resolvents at $i$ and $-i$ converge strongly. Products preserve strong convergence for uniformly bounded nets; uniform approximation by the algebra of these resolvent functions then gives the displayed statement for every continuous function vanishing at infinity.

For a normal operator with disconnected spectrum $K=K_1\bigsqcup K_2$, the indicator function of a nonempty proper clopen component union is continuous on $K$. The continuous functional calculus makes it a nonzero proper self-adjoint linear projection commuting with $T,T^{\ast }$. Its range is therefore a nontrivial closed reducing subspace. No eigenvector is needed.