Spectrum of Star of Element in *-Algebra

Theorem

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

Let $a \in A$.

Let $\sigma_A$ denote the spectrum.

Then:

$\map {\sigma_A} {a^\ast} = \set {\overline \lambda : \lambda \in \map {\sigma_A} a}$

Proof

From the definition of an involution, we have:

$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\mathbf 1}_A^\ast - a^\ast$

From Identity Element in Unital *-Algebra is Hermitian, we therefore have:

$\paren {\lambda {\mathbf 1}_A - a}^\ast = \overline \lambda {\mathbf 1}_A - a^\ast$

From Inverse of Star of Element in Unital *-Algebra: Corollary, we deduce that:

$\lambda {\mathbf 1}_A - a$ is invertible if and only if $\overline \lambda {\mathbf 1}_A - a^\ast$ is invertible.

That is to say:

$\lambda \in \map {\sigma_A} a$ if and only if $\overline \lambda \in \map {\sigma_A} {a^\ast}$.

We conclude:

$\map {\sigma_A} {a^\ast} = \set {\overline \lambda : \lambda \in \map {\sigma_A} a}$

$\blacksquare$

Sources

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