36.13 Quotients of $C^{*}$-Algebras
Lemma 36.13.1.label Let $A$ be a unital $C^{*}$-algebra and $I \subset A$ be a closed two-sided ideal, then for each $x \in I$, $x^{*} \in I$ as well.
Proof. By Theorem 36.12.5, there exists an increasing net $\angles{e_\beta}_{\beta \in B}\subset I \cap \ol{B_A(0, 1)}$ such that $e_{\beta} x \to x$ and $xe_{\beta} \to x$ for all $x \in I$. As $A$ is a $C^{*}$-algebra,
Given that $I$ is two-sided, $x^{*}e_{\beta} \in I$ for all $\beta \in B$. Since $I$ is closed, the above implies that $x^{*} \in I$ as well.$\square$
Lemma 36.13.2.label Let $A$ be a unital $C^{*}$-algebra, $I \subset A$ be a closed two-sided ideal, and $\angles{e_\beta}_{\beta \in B}$ be an increasing approximate identity for $I$, then for each $x \in A$,
Proof, [Lemma 15.6, Zhu93]. For each $y \in I$, $ye_{\beta} \to y$. Thus
As the above holds for all $y \in I$, $\limsup_{\beta \in B}\norm{xe_\beta - x}_{A} \le \norm{x + I}_{A/I}$.
On the other hand, since $I$ is a two-sided ideal, $xe_{\beta} \in I$ for all $\beta \in B$. Thus $\liminf_{\beta \in B}\norm{xe_\beta - x}_{A} \ge \norm{x + I}_{A/I}$.$\square$
Theorem 36.13.3.label Let $A$ be a unital $C^{*}$-algebra and $I \subset A$ be a closed two-sided ideal, then $A/I$ equipped with the quotient norm is a $C^{*}$-algebra.
Proof. Under the quotient structures, $A/I$ is an involutive Banach algebra. It remains to show that $\norm{x + I}_{A/I}^{2} = \norm{x^*x + I}_{A/I}$ for all $x \in A$.
By Lemma 36.13.1, $I$ is a $C^{*}$-algebra. Thus Theorem 36.12.5 implies the existence of an increasing approximate identity $\angles{e_\beta}_{\beta \in B}\subset I$ for $I$.
For each $x \in A$ and $y \in I$,
As the above holds for all $y \in I$, $\norm{x + I}_{A/I}^{2} \le \norm{x^*x + I}_{A/I}$. Thus $A/I$ equipped with the quotient norm is a $C^{*}$-algebra.$\square$
Post a Comment