Definition:Continuous Functional Calculus/Non-Unital Algebra

Definition

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:

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

Also see

• Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal/Corollary shows that $\map f {\tuple {x, 0} } \in A_0$, hence there exists such a $\map f x \in A$
• Continuous Functional Calculus in Non-Unital Algebra is consistent with that in Unital Algebra shows that if $A$ is unital, the $\map f x$ thus obtained is the same as that obtained from the continuous functional calculus for $A$.