K-theory of AF-algebras from braided C*-tensor categories

Abstract

We associate to each Temperley-Lieb-Jones C*-tensor category TLJ pδq with parameter δ in the discrete range t2 cospπ{pk`2qq : k " 1, 2, . . .u Y t2u a certain C*-algebra B of compact operators. We use the unitary braiding on TLJ pδq to equip the category Mod B of (right) Hilbert B-modules with the structure of a braided C*-tensor category. We show that TLJ pδq is equivalent, as a braided C*-tensor category, to the full subcategory Mod f B of Mod B whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.

Realizing the Braided Temperley–Lieb–Jones C*-Tensor Categories as Hilbert C*-Modules

“…[15][16][17]) describing the fusion ring of the category of level k representations of the loop group LSU (2) in terms of twisted equivariant K -theory. Related to this, we observed in [1] that the K 0group of certain approximately finite-dimensional (AF) C*-algebras has a ring structure that is closely related to the fusion ring of TLJ (δ). For example, the K 0group of the inductive limit TLJ ∞ (δ) = lim n TLJ n (δ) of Temperley-Lieb-Jones C*algebras, whose Bratteli diagram is given in [30], is a localization of the fusion ring of TLJ (δ).…”

“…Given a (small) rigid C*-tensor category C, Yuan in [57] constructed a unital C*-algebra A and a fully faithful monoidal *-functor from C into the category A Mod A of finite type Hilbert C*-bimodules over A, the tensor product in A Mod A being given by interior tensor product. A variant of Yuan's construction yields a fully faithful monoidal *-functor from TLJ (δ) into A Mod A , where A is the unital AF-algebra whose Bratteli diagram arises from the fusion graph of f (0) ⊕ f (1) (in the notation of Sect. In the present paper, we make use of Yuan's formalism in defining certain Hilbert spaces and bounded operators.…”

“…[53]). If δ ≥ 2 then the Jones-Wenzl projections form an infinite sequence ( f (n) ) ∞ n=0 with f (n) ∈ TLJ n (δ) for all n, where f (0) is the empty diagram and f (1) is a single strand. The remaining Jones-Wenzl projections are defined via Wenzl's recursive formula (see e.g.…”

“…equation (21) in [39], in which δ is equal to q +q −1 in their notation). It is a fact that f (1) n+1) in TLJ (δ) for all n ≥ 1. If δ = 2 cos(π/(k + 2)) with k ≥ 1 then the Jones-Wenzl projections form a finite sequence f (0) , f (1) , .…”

“…In either case, the category TLJ (δ) is generated by the object π = f (1) in the sense that every simple object occurs as a direct summand of some tensor power π ⊗n of π . Note in this connection that Hom(π ⊗n , π ⊗m ) = TLJ m,n (δ) for all n, m ≥ 0.…”