Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-10-29 , DOI: 10.1142/s021919972050073x
Daniele Mundici

An AF algebra $𝔄$ is said to be an AF$ℓ$ algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF$ℓ$ algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF$ℓ$ algebra $𝔄$, generates a Bratteli diagram of $𝔄.$ We generalize this result to the case when $𝔄$ has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

down
wechat
bug