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This is actually something in a paper but the author claimed it without proof. Let x be a positive elment of norm 1 in a $C^*-$algebra A, and Her(x) is the hereditary subalegbra generated by x. Given $\epsilon>0$,let $f_\epsilon$ be thecontinuous function on R defined as follow:

$f_\epsilon \equiv 0 \quad on \quad [-\infty,\epsilon/2]$

$f_\epsilon \quad is \quad linear \quad on\quad [\epsilon/2,\epsilon]$

$f_\epsilon \equiv 0 1 \quad on\quad [\epsilon, +\infty]$

So $f_\epsilon$ increase to the identiy function on [0,1] when $\epsilon$ decrease to 0, and $Her(x)=\overline{\cup_{\epsilon>0} f_\epsilon(x)Af_\epsilon(x)}$. Let p be a projection in Her(x). then how do we know that there must exist a $\epsilon$ such that $p\in \overline{f_\epsilon(x)Af_\epsilon(x)}$? Or more generally, Let A be the inductive limit of {$A_n$}, and p is a projection in A,does it follow that p is actually in some $A_n$?

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\documentclass{paper} \begin{document}

This is actually something in a paper but the author claimed it without proof. Let x be a positive elment of norm 1 in a $C^*-$algebra A, and Her(x) is the hereditary subalegbra generated by x. Given $\epsilon>0$,let $f_\epsilon$ be thecontinuous function on R defined as follow:\ \begin{math} \quad \quad follow:

$f_\epsilon \equiv 0 \quad on \quad [-\infty,\epsilon/2] \ \quad \quad -\infty,\epsilon/2]$

$f_\epsilon \quad is \; linear\;on\quad [\epsilon/2,\epsilon] \ \quad linear \quad on\quad [\epsilon/2,\epsilon]$

$f_\epsilon \equiv 0 \quad on\quad [\epsilon, +\infty] \ \end{math} \infty]$

So $f_\epsilon$ tends increase to the identiy function on [0,1] when $\epsilon$ tends decrease to 0. So , and $Her(x)=\overline{\cup_{\epsilon>0} f_\epsilon(x)Af_\epsilon(x)}$. Let p be a projection in Her(x). then how do we know that there must exist a $\epsilon$ such that $p\in \overline{f_\epsilon(x)Af_\epsilon(x)}$? Or more generally, Let A be the inductive limit of ${A_n}$, {$A_n$}, and p is a projection in A,does it follow that p is actually in some $A_n$?\end{document}

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# Projection in Hereditary C* subalgebra

\documentclass{paper} \begin{document} This is actually something in a paper but the author claimed it without proof. Let x be a positive elment in a $C^*-$algebra A, and Her(x) is the hereditary subalegbra generated by x. Given $\epsilon>0$,let $f_\epsilon$ be thecontinuous function on R defined as follow:\ \begin{math} \quad \quad f_\epsilon \equiv 0 \quad on \quad [-\infty,\epsilon/2] \ \quad \quad f_\epsilon \quad is \; linear\;on\quad [\epsilon/2,\epsilon] \ \quad \quad f_\epsilon \equiv 0 \quad on\quad [\epsilon, +\infty] \ \end{math} So $f_\epsilon$ tends to the identiy function on [0,1] when $\epsilon$ tends to 0. So $Her(x)=\overline{\cup_{\epsilon>0} f_\epsilon(x)Af_\epsilon(x)}$. Let p be a projection in Her(x). then how do we know that there must exist a $\epsilon$ such that $p\in \overline{f_\epsilon(x)Af_\epsilon(x)}$? Or more generally, Let A be the inductive limit of ${A_n}$, and p is a projection in A,does it follow that p is actually in some $A_n$? \end{document}