Positive Linear Functional on C*-Algebra induces Semi-Inner Product

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Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.

Let $f : A \to \C$ be a positive linear functional.

Define $\innerprod \cdot \cdot : A^2 \to A$ by:

$\innerprod x y = \map f {y^\ast x}$

for each $x, y \in A$.


Then $\innerprod \cdot \cdot$ is a semi-inner product on $A$.


Corollary

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.

Let $f : A \to \C$ be a positive linear functional.

Let $x, y \in A$.


Then:

$\cmod {\map f {y^\ast x} }^2 \le \map f {y^\ast y} \map f {x^\ast x}$


Proof

Proof of Conjugate Symmetry

Let $x, y \in A$ we have:

\(\ds \overline {\innerprod x y}\) \(=\) \(\ds \overline {\map f {y^\ast x} }\)
\(\ds \) \(=\) \(\ds \map f {\paren {y^\ast x}^\ast}\) Positive Linear Functional on C*-Algebra preserves Star
\(\ds \) \(=\) \(\ds \map f {x^\ast y}\) $(\text C^\ast 4)$ and $(\text C^\ast 1)$ in Definition of Involution on Algebra
\(\ds \) \(=\) \(\ds \innerprod y x\)

$\Box$


Proof of linearity

Let $x, y, z \in A$ and $\alpha \in \C$.

Then we have:

\(\ds \innerprod {x + \alpha y} z\) \(=\) \(\ds \map f {z^\ast \paren {x + \alpha y} }\)
\(\ds \) \(=\) \(\ds \map f {z^\ast x + \alpha z^\ast y}\)
\(\ds \) \(=\) \(\ds \map f {z^\ast x} + \alpha \map f {z^\ast y}\) from the linearity of $f$
\(\ds \) \(=\) \(\ds \innerprod x z + \alpha \innerprod y z\)

$\Box$


Proof of non-negative definiteness

Let $x \in A$.

Then we have:

$\innerprod x x = \map f {x^\ast x}$

From Product of Element of C*-Algebra with its Star is Positive, $x^\ast x$ is positive.

So $\map f {x^\ast x} \in \R_{\ge 0}$.

Hence $\innerprod x x \in \R_{\ge 0}$.

$\blacksquare$


Sources

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