34.2 Subalgebras
Definition 34.2.1 (Generated Subalgebra).label Let $A$ be a unital $C^{*}$-algebra and $S \subset A$, then $A[S]$ is the smallest $C^{*}$-subalgebra of $A$ containing $1$ and $S$.
Proposition 34.2.2.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$, and $\mathcal{S}= S \cup \bracs{x^*|x \in S}$, then
- (1)
The linear span
\[\text{span}\bracs{\prod_{j = 1}^n x_j \bigg | \seqf{x_j} \subset \mathcal{S}}\]is dense in $A[S]$.
- (2)
If for any $x, y \in \mathcal{S}$, $xy = yx$, then $A[S]$ is commutative.
- (3)
For any normal element $x \in A$, $A[x]$ is commutative.
Proposition 34.2.3.label Let $A$ be a unital $C^{*}$-algebra, $B \subset A$ be a $C^{*}$-subalgebra containing $1$, then $G(B) = G(A) \cap B$.
Proof. Let $x \in G(A) \cap B$, then $x^{*}x \in G(A) \cap B$ as well. In particular, $0 \not\in \sigma_{A}(x^{*}x)$. Since $x^{*}x \in A_{sa}$, $\sigma_{A}(x^{*}x) \subset \real$ by Proposition 34.4.6. By Proposition 33.5.12, $\partial \sigma_{B}(x^{*}x) \subset \sigma_{A}(x^{*}x) \subset \real$. Thus $\sigma_{B}(x^{*}x) \subset \real$ as well, which means that $\partial \sigma_{B}(x^{*}x) = \sigma_{B}(x^{*}x) = \sigma_{A}(x^{*}x)$. Therefore $0 \not\in \sigma_{A}(x^{*}x) = \sigma_{B}(x^{*}x)$, $x^{*}x \in G(B)$, and $x \in G(B)$.$\square$
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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