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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 $N$ ?

user540663
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1 Answers1

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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.