Proposition 30.1.3.label Let $A$ be a $C^{*}$ algebra, then:
- (1)
For each $x \in A$, $\norm{x}_{A} = \normn{x^*}_{A}\norm{x}_{A}$.
If $A$ is unital, then
- (2)
For each $\lambda \in \complex$, $\lambda^{*} = \ol \lambda$.
- (3)
For any $x \in A$, $x \in G(A)$ if and only if $x^{*} \in G(A)$.
- (4)
For every $x \in A$, $\sigma_{A}(x^{*}) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$.
- (5)
For each $x \in A$, $[x]_{sp}= [x^{*}]_{sp}$.
Proof. (1): For each $x \in A$, $\norm{x}_{A}^{2} = \normn{x^*x}_{A} \le \norm{x}_{A} \normn{x^*}_{A}$.
(2): For every $x \in A$, $1^{*}x^{*} = (x1)^{*} = x^{*} = (1x)^{*} = x^{*}1^{*}$, so $1^{*} = 1$ by uniqueness of the inverse.
(3): For any $x \in A$, $(x^{-1})^{*}x^{*} = (x^{-1}x)^{*} = 1 = (xx^{-1})^{*} = x^{*}(x^{-1})^{*}$.$\square$
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