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:

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

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

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

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 $A_n$?

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:

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

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

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

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 $A_n$?

\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}

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:

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

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

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

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 $A_n$?