If $A$ is a unital direct limit of C*- algebras, why can we assume that the connecting maps are unital?

Question

I know that we may assume that each $\phi _{n}$ is injective. Then how to show that we may assume $\phi_n$ is unit preserving when $n \geq N$ for some $N$ ?

Answer 1

If all connecting maps are injective you can do the following:

• First show that almost all $A_n$ are unital. This can be done by approximating $1_A$ by positive contractions and applying functional calculus to get projections that are close to $1_A$.
• Then use that an invertible projection is already the unit.
• Show that the connecting maps preserve the units you just have found.