Product of Element in *-Star Algebra with its Star is Hermitian

Theorem

Let $\struct {A, \ast}$ be a $\ast$-algebra.

Let $a \in A$.

Then $a^\ast a$ and $a a^\ast$ are Hermitian.

Proof

We have:

\(\ds \paren {a^\ast a}^\ast\)

\(=\)

\(\ds a^\ast \paren {a^\ast}^\ast\)

$(\text C^\ast 3)$

\(\ds \)

\(=\)

\(\ds a^\ast a\)

$(\text C^\ast 1)$

and:

\(\ds \paren {a a^\ast}^\ast\)

\(=\)

\(\ds \paren {a^\ast}^\ast a^\ast\)

$(\text C^\ast 3)$

\(\ds \)

\(=\)

\(\ds a a^\ast\)

$(\text C^\ast 1)$

So $a^\ast a$ and $a a^\ast$ are both Hermitian.

$\blacksquare$

Sources

• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.1$: $C^\ast$-Algebras