Corollary 34.2.4.label Let $A$ be a unital $C^{*}$-algebra, $B \subset A$ be a $C^{*}$-subalgebra containing $1$, and $x \in B$, then $\sigma_{A}(x) = \sigma_{B}(x)$.
Proof. By Proposition 34.2.3.$\square$
Corollary 34.2.4.label Let $A$ be a unital $C^{*}$-algebra, $B \subset A$ be a $C^{*}$-subalgebra containing $1$, and $x \in B$, then $\sigma_{A}(x) = \sigma_{B}(x)$.
Proof. By Proposition 34.2.3.$\square$
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