Definition:Continuous Functional Calculus

Definition

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.

Let $x \in A$ be normal.

Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.

Let $\iota : \map {\sigma_A} x \to \C$ be the inclusion mapping.

Let $\Theta_x : \map \CC {\map {\sigma_A} x} \to A$ be a unital $\ast$-algebra homomorphism such that:

$\map {\Theta_x} \iota = x$

We call $\Theta_x$ the continuous functional calculus of $x$.

For $f \in \map \CC {\map {\sigma_A} x}$, we define:

$\map f x = \map {\Theta_x} f$

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra that is not necessarily unital.

Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the unitization of $\struct {A, \ast, \norm {\, \cdot \,} }$.

Let:

$A_0 = \set {\tuple {a, 0} : a \in A}$

Let $x \in A$ be normal.

Let $\map {\sigma_A} x$ denote the spectrum of $x$ in $A$.

Let $\map {\sigma_{A_+} } {\tuple {x, 0} }$ denote the spectrum of $\tuple {x, 0}$ in $A_+$.

Let $f : \map {\sigma_{A_+} } {\tuple {x, 0} } \to \C$ be continuous with $\map f 0 = 0$.

Let $\map f {\tuple {x, 0} }$ be obtained from the continuous functional calculus for $A_+$, with $f$.

We take $\map f x \in A$ to be such that:

$\map f {\tuple {x, 0} } = \tuple {\map f x, 0}$

Existence and Uniqueness of Continuous Functional Calculus shows the existence and uniqueness of $\Theta_x$ for each normal $x \in A$, establishing that $\map f x$ is well-defined