Proving that $D$ is a directed set.

Question

Let $A$ be a $C^{\ast}$-algebra and $D$ be the set of all positive elements in $A$ with norm strictly less than $1.$ Let $a,b \in D$ and $a_1 = a (1 - a)^{-1},$ $b_1 = b (1 - b)^{-1},$ $c = (a_1 + b_1) (a_1 + b_1 + 1)^{-1}.$ Show that $c \in D$ and $a \leq c, b \leq c.$

I find it difficult to show this result. Any help in this regard would be greatly appreciated.

Thanks for your time.

EDIT $:$ I am getting stuck in proving that $a \leq c$ and $b \leq c.$ Could anyone please help me in this regard?

Answer 1

Since you showed $c\in{D}$, notice: $$1+a_{1}\le1+a_{1}+b_{1}$$ $$\left(1+a_{1}\right)^{-1}\ge\left(1+a_{1}+b_{1}\right)^{-1}$$ $$1-\left(1+a_{1}\right)^{-1}\le1-\left(1+a_{1}+b_{1}\right)^{-1}$$ Check that lhs is $a$ and rhs is $c$. Same works for $b$.