I have currently been working with full $\ast$-homomorphisms and have been in need of deriving the following fact concerning these. First of all, let $A$ and $B$ be unital C$^\ast$-algebras. A unital $\ast$-homomorphism $\gamma \colon A \longrightarrow B$ is called $\textit{full}$ if $$ I(a):=\bigg \lbrace \sum_{i=1}^n x_i \gamma(a) y_i : x_i, y_i \in B, \ n\in \mathbb{N} \bigg \rbrace $$
is norm dense in $B$ for each nonzero $a\in A$. Following exercise $4.8$ in the book "An introduction to K-theory for C$^\ast$-algebras" by Rørdam, Larsen and Lausten, I am trying to prove that if $\gamma$ is full, then for every nonzero positive element $a\in A$ there exist an $n\in \mathbb{N}$ and $x_1, x_2, \ldots , x_n \in A$ fulfilling $$ 1_B = \sum_{i=1}^n x_i a x_i^* $$
During the proof, I need the following fact, which eludes me: Suppose $a\geq 0$ in $A$ and $1_A \leq a$. Then there exists some $r\in A$ such that $1_A = rar^*$. Any hints would be greatly appreciated! This is exercise $4.8$ part (iii) to be precise.
Since $a\geq 1_A$, then $a$ is invertible. So you can take $r=a^{-1/2}$.