Corollary 34.3.5.label Let $A$ be a unital $C^{*}$-algebra over $\complex$, then for each $x \in A$,
\[\norm{x}_{A}^{2} = \sup\bracs{|\lambda|\ | \lambda \in \sigma_A(x^*x)}\]
In particular, there exists at most one norm on $A$ making it a $C^{*}$-algebra.
Proof. Since $x^{*}x$ is self-adjoint, Theorem 34.3.3 implies that
\[\norm{x}_{A}^{2} = \norm{x^*x}_{A} = \sup\bracs{|\lambda|\ | \lambda \in \sigma_A(x^*x)}\]
which depends only on the algebraic structure of $A$.$\square$
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