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\begin{proposition}
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\label{proposition:c-star-algebra-gymnastics}
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Let $A$ be a $C^*$ algebra, then:
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Let $A$ be a $C^*$-algebra, then:
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\begin{enumerate}
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\item For each $x \in A$, $\norm{x}_A = \normn{x^*}_A\norm{x}_A$.
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\item For each $x \in A$, $\norm{x}_A^2 = \normn{x^*}_A\norm{x}_A$.
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\end{enumerate}
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If $A$ is unital, then
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