Corollary 34.7.5.label Let $A$ be a unital $C^{*}$-algebra, then
- (1)
For any $x \in A_{sa}$, there exists a unitary element $u \in A$ such that $x = (u + u^{*})/(2\norm{x}_{A})$.
- (2)
For any $x \in A$, there exists unitary elements $u, v \in A$ such that $x = (u + u^{*})/(\norm{x}_{A}) + i(v + v^{*})/(2\norm{x}_{A})$.
Proof. (1): Assume without loss of generality that $\norm{x}_{A} \le 1$. In which case, $\sigma(x) \subset [-1, 1]$, and $f(\lambda) = \lambda + i\sqrt{1 - \lambda^{2}}$ is defined and continuous on $[-1, 1]$. Furthermore, $|f(\lambda)| = 1$ for all $\lambda \in [-1, 1]$. Thus Corollary 34.7.4 implies that $f(x)$ is unitary. Finally, since $f + \ol f = 2\text{Id}$, $x = (f(x) + \ol{f(x)})/(2\norm{x}_{A})$.$\square$
Post a Comment