Remark

Let A be a unital involutive algebra and let $\widetilde{A}$ be the unital involutive algebra deduced from A by adjunction of a unit element $e$. For every $x\in A$, one has Sp$'_A(x) =$ Sp$_A(x)\cup \{0\}$. Let $x$ be a normal element of A. Then it is a normal element of $\widetilde{A}$, and therefore one has two mappings of functional calculus in A, the first defined on $\mathscr{C}$ (Sp$_A(x)$) and the second on $\mathscr{C}'$(Sp$'_A(x)$). Let $f'\in \mathscr{C}'$(Sp$'_A(x)$); if $f$ denotes its restriction to Sp$_A(x)$, one then has $f'(x) =f(x)$.