Spectrum of Element in Unital Subalgebra

Theorem
Let $A$ be an algebra over $\C$.

Let $B$ be a unital subalgebra of $A$.

Let $x \in B$.

Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.

Then:
 * $\map {\sigma_A} x \subseteq \map {\sigma_B} x$

Proof
Let $\map G A$ and $\map G B$ be the group of units of $A$ and $B$ respectively.

From Group of Units of Submonoid is Subgroup, we have:
 * $\map G B \subseteq \map G A$

From Set Complement inverts Subsets, we have:
 * $\C \setminus \map G A \subseteq \C \setminus \map G B$

Then, we have: