scholarly journals Some model-theoretic correspondences between dimension groups and AF algebras

2011 ◽  
Vol 162 (9) ◽  
pp. 755-785 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Dimension Groups ◽  
Af Algebras

DIMENSION GROUPS, EMBEDDINGS AND PRESENTATIONS OF AF ALGEBRAS ASSOCIATED TO SOLVABLE LATTICE MODELS

1989 ◽  
Vol 04 (20) ◽  
pp. 1883-1890 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Lattice Models ◽  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. Embeddings A(Γ1)→A(Γ2) of non-commutative AF algebras associated with graphs Γ1 and Γ2 are discussed from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ))≃ ℤ[t], with positive cone identified with {0}∪{P∈ℤ(t): P(λ)>0, λ∈(0,γ]}, where γ=γ(Γ)= ||Γ||−2<1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||. For certain finite T-shaped graphs K0(A(Γ))≃ℤ[t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ, and positive cone identified with {0}∪{f+<Q>: f(γ)>0} where γ=γ(Γ)=||Γ||−2. Hence there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||, and Q(Γ1) divides Q(Γ2). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory. Moreover, for these T-shaped graphs there is an algebraic presentation which further illuminates the above embeddings. This presentation involves a new projection and a new relation in addition to those of Temperley-Lieb, and gives a rigidity above index four.

scholarly journals Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams

2000 ◽  
Vol 20 (6) ◽  
pp. 1639-1656 ◽  
Author(s):  
OLA BRATTELI ◽  
PALLE E. T. JØRGENSEN ◽  
KI HANG KIM ◽  
FRED ROUSH
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Negative Integer ◽  
Matrix Elements ◽  
Integer Matrix ◽  
Incidence Matrices ◽  
Bratteli Diagrams ◽  
Dimension Groups ◽  
Necessary And Sufficient ◽  
Af Algebras

We first study situations where the stable AF algebras defined by two square primitive non-singular incidence matrices with non-negative integer matrix elements are isomorphic, even though no powers of the associated automorphisms of thecorresponding dimension groups are isomorphic. More generally we consider necessary and sufficient conditions for two such matrices to determine isomorphic dimension groups.We give several examples.

DIMENSION GROUPS AND EMBEDDINGS OF GRAPH ALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 291-327 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Graph Algebras ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. We discuss embeddings A(Γ1) → A(Γ2) of AF algebras associated with graphs Γ1 and Γ2 from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ)) ≅ ℤ [t], with positive cone identified with {0}∪ {P ∈ ℤ [t]: P (λ) > 0, λ ∈ (0, γ]}, where γ = γ (Γ) =||Γ||−2 < 1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| ≤ ||Γ2||. For certain finite T-shaped graphs K0 (A(Γ)) ≅ ℤ [t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ and positive cone identified with {0}∪ {f + <Q>: f(γ) > 0} where γ = γ(Γ) = ||Γ||-2. Hence there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| = ||Γ2||, and Q(Γ2) divides Q(Γ1). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory.

Classification of AF-Algebras

2010 ◽  
pp. 109-132
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
Af Algebras

Quasidiagonal extensions and $AF$ algebras

1998 ◽  
Vol 311 (2) ◽  
pp. 233-249 ◽  
Author(s):  
S&#x000F8;ren Eilers ◽  
Terry A. Loring ◽  
Gert K. Pedersen
Keyword(s):  
Af Algebras

An AF Algebra Associated with the Farey Tessellation

2008 ◽  
Vol 60 (5) ◽  
pp. 975-1000 ◽  
Author(s):  
Florin P. Boca
Keyword(s):  
Half Plane ◽  
Path Algebra ◽  
Cutting Sequences ◽  
Upper Half Plane ◽  
Af Algebras ◽  
Algebra Model

AbstractWe associate with the Farey tessellation of the upper half-plane an AF algebra encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen AF algebras arise as quotients of . Using the path algebra model for AF algebras we construct, for each τ ∈ ( 0, ¼], projections (En) in such that EnEn±1En ≤ τ En.

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