Armitage is our Danny Ocean. He's the guy putting this whole heist together. The only problem is he technically doesn't exist. Oops.
Let's back up a bit. In the novel's backstory, there was a war (the specifics don't matter much), and in this war, there was an operation codenamed Screaming Fist. U.S. Special Forces tried to disrupt some high level Russian computers. Things went bad and everyone died except a guy named Colonel Willis Corto. Corto was severely wounded and then betrayed by his higher ups. The betrayal caused Corot to eventually go insane, and an experimental treatment was tested on him. The idea was to graft a brand new personality over the broken one, creating essentially an entirely new person. That person's name was Armitage.
What a ride. Still with us? Cool.
Like Molly and Case, Armitage is a man burdened by the past. On the one hand, Wintermute created the Armitage personality to serve as a Band-Aid for the mental wound that is Corto. He lives only to do Wintermute's bidding. When not wheeling and dealing for Wintermute, he "[s]its and stares at the wall" (7.90). As things are, his past is Wintermute, and he has no future beyond the AI.
On the other hand, Corto—the Corto that's still trapped inside Armitage, hidden beneath the layers of Wintermute loyalty—is obsessed with his past and way he was betrayed. The obsession is so strong that it breaks him mentally, causing schizophrenia and insanity, and making him ripe for cybernetic brainwashing.
In this way, Armitage becomes a foil for Molly and Case. Molly and Case both try to deal with their pasts in their own ways, but Corto-turned-Armitage can only do one of two things. As Armitage, he can ignore it. As Corto, he can relive it.
Neither seems to work out to well for him. As the story progresses, the old Corto starts to bust through the Armitage façade. And the results are, shall we say, not good. It's yet another example of characters in the novel striving to move beyond their pasts, and coming up short.