2 Order Lemmas
Let \(\mathcal{A}\) be a unital \(C^{*}\)–algebra in this section. We collect lemmas for the ordering of elements in \(\mathcal{A}\). Recall that a selfadjoint element \(a \in \mathcal{A}\) is said to be positive if its spectrum is contained in \(\mathbb {R}_{\ge 0}\). This is written \(a\ge 0\). If \(a,b \in \mathcal{A}\) are selfadjoint then \(b \le a\) if \(b-a \ge 0\).
(Sak 1.4.4) Let \(h \in \mathcal{A}\). TFAE:
\(h \ge 0\);
There exists \(x\in \mathcal{A}\) such that \(h=x^{*}x\).
Proof
If \(h,a \in \mathcal{A}\) with \(h\ge 0\) then \(a^{*}ha\ge 0\)
Proof
By Lemma 27, \(a^{*}ha=a^{*}x^{*}xa=(xa)^{*}(xa)\ge 0\).
If \(x\in \mathcal{A}\) is selfadjoint then \(\| x\| 1-x \ge 0\).
Proof