Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $A$ be a unital $C*$ algebra. $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertable. I proved one direction : if $a$ is invertable, then $a$ is strictly positive , but I have no idea how to prove the other direction.
Thank you

share|cite|improve this question
up vote 3 down vote accepted

Since $\overline{aAa}=A$, there is $x\in A$ such that $\|axa-1\|<1/2$. Therefore $axa$ is invertible. This implies that $a$ has left and right inverse and consequently it is invertible.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.