Stably projectionless Fraïssé limits

Abstract

We realise the algebra W, the algebra Z 0 and the algebras Z 0 ⊗ A, where A is a unital UHF algebra as Fraïssé limits of suitable classes of structures. In doing so, we show that such algebras are generic objects without the use of any classification result.

Metrics on trace spaces

“…Let A ⊆ C([0, 1], M n ) and B ⊆ C([0, 1], M m ) be one dimensional NCCW complexes. Following [19], we call a * -homomorphism ϕ : A → B diagonal if m = ln for some l ∈ N, and there are continuous maps ξ 1 , . .…”

“…There is a unique simple inductive limit of Razak blocks W = lim − → (A i , α i ) that has a unique bounded trace (see [16]). We will use the 'generic' construction of W in [19,Definition 3.10].…”

“…By having moved far enough down the sequence lim − → (A i ⊗ C i , α i ⊗ β i ), we can ensure that the infimum condition holds for ϕ for some δ 1 . By universality of the construction of W in [19], for all sufficiently large i, there are diagonal embeddings ϕ ′ j of A n,k into A i ⊗ M Nj for 1 ≤ j ≤ m such that λ • ϕ ′ j ∈ T g (A n,k ) for every j (as in [19,Proposition 3.5]) and such that the infimum condition also holds for ϕ ′ = ϕ ′ 1 ⊕ • • • ⊕ ϕ ′ m and some δ 2 .…”