Lemma 34.10.2.label Let $A$ be a unital $C^{*}$-algebra and $\phi \in A^{*}$ be a positive linear functional, then the mapping
\[A \times A \to \complex \quad (x, y) \mapsto \dpn{x, y}{\phi}:= \dpn{y^*x, \phi}{A}\]
is a pseudo inner product. In particular, for any $x, y \in A$,
\[|\dpn{y^*x, \phi}{A}|^{2} = |\dpn{x, y}{\phi}|^{2} \le \dpn{x, x}{\phi}\cdot \dpn{y, y}{\phi}\]
Proof. By the Cauchy-Schwarz inequality.$\square$
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