Invariants for triangular 𝐴𝐹 algebras

Classification of limits of triangular matrix algebras

“…A number of recent papers [1,4,5,6,8,9,10,12,13,14] have been devoted to non-self-adjoint subalgebras of UHF algebras (and, more generally, of AF C*-algebras). Usually, the algebras in question contain a canonical masa in the sense of Stratila and Voiculescu [11]; often they are triangular: the intersection with the adjoint algebra is the canonical masa.…”

“…The second family arises from refinement embeddings with a twist (cf. [5]), and the third family is a class of homogeneous nest subalgebras related to fractal subsets of the unit square.…”

“…Note that a, = AdU n op t for a suitable unitary operator l/ n induced by a permutation n of the minimal projections in appeared in [1,5,8], while classification of the corresponding family with p in place of a appeared in [5,8]. For each family, the supernatural number associated with the sequence k l ,k 2 ,k 3 ,... is a complete invariant, i.e.…”

A Note on Subdiagonality for Triangular AF Algebras

“…J. Peters, Y. Poon, and B. Wagner have introduced two different notions of maximality for triangular AF algebras (cf. [PPW,Proposition 2.24. and Example 3.25.]). One of them is strong maximality, which we now recall.…”

Algebraic orders and chordal limit algebras

“…However, putting an order on the scale describes the non-selfadjoint part. Following [13], define the diagonal order on projections by saying p is less than q if there is a partial isometry, w, that normalises the diagonal of the algebras and so that w'w = q and ww' = p. This induces a well-defined ordering, S(A), on the scale of the K 0 -group, called the algebraic order [15]. In [15] Power also considered a second order, the strong algebraic order S\(A), where an additional condition is imposed on the partial isometries w: namely, conjugation by w preserves the diagonal ordering on projections.…”