# Existence of a approximate unit $U_{n}^{2}$ for a $C^{*}$-algebra $if${U_{n}}$is an approximate unit for a$C^{*}$-algebra A. Is${U_{n}^{2}}$is an approximate unit for a$C^{*}$-algebra A? Thank in advance. ## 2 Answers Let$\{U_n\}$be your approximate unit. The$U_n$are positive so that the$U_n^2=U_n^*U_n$are also positive. Also by the C*-equation we have $$\|U_n^2\|=\|U^*_nU_n\|=\|U_n\|^2\leq 1\Rightarrow \|U_n^2\|\leq 1.$$ You should be able to show/find that for any$a\in A$$$\|U_naU_n-a\|\rightarrow 0\qquad(*)$$ and any$x\in Awe have $$\|U_nx-xU_n\|\rightarrow 0.\qquad(**)$$ Note \begin{align}\|U_n^2a-a\|&\leq \left\|U_n^2 a-U_naU_n\right\|+\|U_naU_n-a\| \\&=\left\|U_n(U_na)-(U_na)U_n\right\|+\|U_naU_n-a\|\rightarrow0. \end{align}. • thanks for your reply and your time. – reza Dec 10 '14 at 20:36 Approximate units are bounded by1\$, so \begin{align} \|U_n^2X-X\|&\leq\|U_n^2X-U_nX\|+\|U_nX-X\|=\|U_n(U_nX-X)\|+\|U_nX-X\|\\ &\leq\|U_nX-X\|+\|U_nX-X\|=2\|U_nX-X\| \end{align}

• thanks. but for increasing the approximate unit U_{N}^{2} Does not hold, unless C*-algebra is commutative,true? – reza Dec 8 '14 at 19:46
• Yes, you are right. I wasn't thinking about the monotonicity property. – Martin Argerami Dec 8 '14 at 21:28