DIMENSION GROUPS AND EMBEDDINGS OF GRAPH ALGEBRAS

Abstract

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.