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An element $x$ in the $C^*$-algebra $A$ is well-supported if there is a $p\in A$ with $x=xp$ and $x^*x$ invertible in $pAp$.

That is the definition, but I cannot catch the key of it. Maybe you can show me some examples, which one is well-supported, which one is not, and what is the motivation.

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    In the special case where $A=B(H)$, you can show that $p$ is the orthogonal projection onto $\ker(x)^\perp$, and $x$ is well-supported if and only if $x|_{\ker(x)^\perp}$ is bounded below.2012-07-03

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