Comparison of positive elements and Hilbert C*-modules

Question

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras.

Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak A$ such that $a\leq b$. Then

1. there is an inclusion of $\overline{a\mathfrak A}$ into $\overline{b\mathfrak A}$ as right Hilbert $\mathfrak A$-modules;
2. $\overline{a\mathfrak A}\equiv\overline{aa^*\mathfrak A}$.

I believe that the idea behind the proofs of these two facts is basically the same (I was thinking in terms of support of spectral projections). So the question is: how do you prove such facts, and similar properties for C*-algebras that are along these lines?

Any reference to existing literature is also welcomed.

Cheers!