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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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Is $p$ a projection? Where did you encouter this definition? –  Rasmus Sep 27 '11 at 11:02
Yes,a projection p∈A,I am sorry I forget it. –  Strongart Sep 29 '11 at 10:20
Could you give some context? Where did you encouter this definition? –  Rasmus Sep 29 '11 at 13:53
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. –  Jonas Meyer Jul 3 '12 at 6:43

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