# Bratteli diagram decided by AF-algebras

In general, an AF-algebra can has some different Bratteli diagrams. We can add some identical arrows to make the Bratteli diagrams different, but it is too trivial, are any good examples?

For which AF-algebras, its Bratteli diagram can be unique decided? I think the UHF algebras should be. How about others?

A same question is at here: https://math.stackexchange.com/questions/654472/bratteli-diagram-decided-by-af-algebras

• I don't quite understand what you call "different" Bratteli diagrams. As you point out, they can "look" different but nevertheless yield the same AF-algebra. There is a way to define an equivalence relation among Bratelli diagrams such that two diagrams are equivalent if and only if the associated AF-algebras are isomorphic. If you're interested in UHF-algebras, there is a way to express the CAR algebra as a (non-trivial) direct limit of algebras of the form $M_{2^n}\oplus M_{2^n}$ (as opposed to the usual $M_{2^n}$). I can give you the details if you are interested. – Eusebio Gardella Jan 31 '14 at 22:17
• By "give you the details" I mean "write a proper answer" and not just a comment. – Eusebio Gardella Jan 31 '14 at 22:18
• I do not find the exact definition for the Bratteli diagram, so we can choose a suitable equivalence, I think the isomorphism as a minimum arrow diagrams is suitable, here minimum means no identical arrows. Maybe you have other natural choise. – Strongart Feb 3 '14 at 11:38