Consider the following fragment in the text "$C^*$-algebras and operator theory by Murphy":
Could someone explain why the marked step is true? I don't see how this follows from $\Vert u(x) \Vert^2 = \Vert u^* u(x) \Vert^2$. Thanks in advance.
2026-03-26 14:33:03.1774535583
Question about proof characterisation partial isometry
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If $\|u(x)\|^2 = \|u^*u(x)\|^2$ for all $x$ then $\|u(1-u^*u)\,(x)\|^2=\|u^*u(1-u^*u)\,(x)\|^2=\|(u^*u-(u^*u)^2)\,(x)\|^2$. But $u^*u$ is a projection so what is left here is $\|u(1-u^*u)\,(x)\|^2=\|(u^*u-u^*u)\,(x)\|^2=0$ for all $x$.