9.7 Order Structures of $C^{*}$-Algebras
Definition 9.7.1 (Positive).label Let $A$ be a $C^{*}$-algebra and $x\in A$, then $x$ is positive if there exists $y\in A$ such that $x=y^{*}y$.
Proposition 9.7.2.label Let $A$ be a unital $C^{*}$-algebra and $x\in A$ be normal, then $x$ is positive if and only if $\spec{A}{x}\suf [0,\infty)$.
Proof. By Proposition 9.5.8 and Definition 9.5.9, $x$ is positive if and only if $\scr{F}_{A[x]}(x)=\text{Id}$ is positive in $C(\spec{A}{x};\com)$ if and only if $\spec{A}{x}=\scr{F}_{A[x]}(x)(\OM(A[x]))\suf [0,\infty)$ by Theorem 9.4.12.$\square$
Post a Comment