Category:Continuous Functional Calculus

This category contains results about Continuous Functional Calculus.
Definitions specific to this category can be found in Definitions/Continuous Functional Calculus.

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:

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

where $\map \CC {\map {\sigma_A} x}$ is the space of continuous functions on $\map {\sigma_A} x$.

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

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

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