I want to show the following statement but I do not know how.
Let $A$ be a unital C*-algebra. Let $a$ be a positive element such that $a\geq 1_A$. Show that there exists $r$ in $A$ with $1_A=rar^*$.
Thanks for all helps!
A question on positive elements bigger than $1_A$ in a unital C*-algebra $A$
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functional-analysis
operator-algebras
1 Answers
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Since $1\leq a$, we know $a-1$ is positive, so $\sigma(a-1)\subset[0,\infty)$, thus $\sigma(a)\subset[1,\infty)$ and $a$ is invertible (here $\sigma(a)$ denotes the spectrum of $a$). Hence its positive square root $a^{1/2}$ is also invertible. Thus, we have $$ 1=1\cdot1=\left(a^{-1/2}a^{1/2}\right)\cdot\left(a^{1/2}a^{-1/2}\right) =a^{-1/2}\cdot\left(a^{1/2}a^{1/2}\right)\cdot a^{-1/2}=a^{-1/2}\cdot a\cdot a^{-1/2} $$ If we let $r=a^{-1/2},$ then $r=r^*$ and $$ 1=rar=rar^*. $$