On the simplicity of twisted k-graph C -algebras

Transcription

1 Preliminary report of work in progress Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS14, Kansas State University, Manhattan, 27 May 2014

2 Introduction C -algebras of higher-rank graphs (or k-graphs) were introduced in [KP00] as generalizations of graph C -algebra which model C -algebras arising from certain group actions on buildings discovered by Robertson and Steger. If a k-graph Λ satisfies mild hypotheses, there is a path groupoid G Λ such that C (Λ) = C (G Λ ). Using this isomorphism it was shown that C (Λ) is simple iff Λ is aperiodic and cofinal. Given a k-graph Λ and a T-valued 2-cocycle c, one may form the twisted k-graph C -algebra C (Λ, c) (see [KPS]). Examples include all noncommutative tori and crossed products of Cuntz algebras by quasifree automorphisms. Our goal is to characterize the simplicity of C (Λ, c). Sufficient conditions were given in (see [SWW]). This talk is based on joint work with David Pask and Aidan Sims of the University of Wollongong.

3 k-graphs Let k N := {0, 1, 2,... }. Definition (see [KP00]) Let Λ be a countable small category and let d : Λ N k be a functor. Then (Λ, d) is a k-graph if it satisfies the factorization property: For every λ Λ and m, n N k such that d(λ) = m + n there exist unique µ, ν Λ such that λ = µν, d(µ) = m and d(ν) = n. Set Λ n := d 1 (n) and identify Λ 0 = Obj (Λ), the set of vertices. An element λ Λ ei is called an edge. We assume throughout that Λ is row-finite and source-free, that is for all v Λ 0, n N k, vλ n := r 1 (v) Λ n is finite and nonempty.

4 Remarks and Examples Let Λ be a k-graph. If k = 0, then d is trivial and Λ is just a set. If k = 1, then Λ is the path category of a directed graph. If k 2, think of Λ as generated by k graphs of different colors that share the same set of vertices Λ 0. The k-graph T k := N k may be regarded as the k-graph analog of a torus. Let Ω k := {(m, n) N k N k m n} be the k-graph with structure maps.... Ω 2 s(m, n) = n r(m, n) = m d(m, n) = n m (l, m)(m, n) = (l, n)

5 The path groupoid The infinite path space Λ is the set of k-graph morphisms x : Ω k Λ. The shift map: for q N k define σ q : Λ Λ by σ q (x)(m, n) = x(m + q, n + q) for Ω k. We define the path groupoid G Λ Λ Z Λ by G Λ := {(x, m n, y) : σ m (x) = σ n (y) for some m, n N k }. The unit space is identified with Λ via the map x (x, 0, x). Λ is cofinal if for every v Λ 0 and x Λ, there are λ Λ and n N k such that s(λ) = x(n, n) and r(λ) = v. If Λ is cofinal, G Λ is minimal. For v Λ 0 the local periodicity group at v P Λ (v) is the set of all m n Z k such that m, n N k and σ m (x) = σ n (x) for all x vλ. Λ is aperiodic if P Λ (v) = 0 for all v Λ 0. If Λ is aperiodic, G Λ is topologically principal, that is, points with trivial isotropy are dense.

6 Categorical Cohomology The categorical cohomology, H cat(λ, A), is the usual cocycle cohomology for groupoids (see [R]) extended to small categories. Consider the 2-cocycle c Z 2 cat(λ, A), that is, a map c : Λ Λ A such that for any composable triple (λ 1, λ 2, λ 3 ) we have c(λ 1, λ 2 ) + c(λ 1 λ 2, λ 3 ) = c(λ 1, λ 2 λ 3 ) + c(λ 2, λ 3 ) and c is a 2-coboundary if there is b : Λ A such that c(λ 1, λ 2 ) = b(λ 1 ) b(λ 1 λ 2 ) + b(λ 2 ). H 2 cat(λ, A) is the quotient group (2-cocycles modulo 2-coboundaries). There is a homomorphism H 2 cat(λ, A) H 2 (G Λ, A) induced by a map c σ c (see [KPS, 6]).

7 The C -algebra C (Λ, c) Definition (see [KPS, 5]) For c Z 2 cat(λ, T) let C (Λ, c) be the universal C*-algebra generated by the set {t λ : λ Λ} satisfying: 1 {t v : v Λ 0 } is a family of orthogonal projections. 2 For λ Λ, t s(λ) = t λ t λ. 3 If s(λ) = r(µ), then t λ t µ = c(λ, µ)t λµ. 4 For v Λ 0, n N k t v = λ vλ n t λ t λ. Remark: If c and c are cohomologous, then C (Λ, c) = C (Λ, c ). Theorem (see [KPS, 7]) Let c Z 2 cat(λ, T) and let σ c Z 2 (G Λ, T) be as above. Then C (Λ, c) = C (G Λ, σ c ).

8 Structure of C (Λ, c) when Λ is cofinal If C (Λ, c) is simple, then Λ is cofinal but not necessarily aperiodic. Suppose Λ is cofinal. Then P Λ := P Λ (v) does not depend on v Λ 0 and there is a short exact sequence of étale groupoids Λ P Λ i G Λ H Λ where H Λ is minimal and topologically principal (see [KPSS]). The cohomology class of i x (σ c ) in H 2 (P Λ, T) is independent of x. Moreover, there is σ Z 2 (G Λ, T) and ω Z 2 (P Λ, T) such that [σ] = [σ c ] and i x (σ) = ω for all x Λ. It follows that C (Λ P Λ, i (σ)) = C 0 (Λ ) C (P Λ, ω). Theorem There is a Fell bundle BΛ c over H Λ such that BΛ c Λ = Λ C (P Λ, ω) and C (Λ, c) = C (H Λ ; BΛ). c

9 Simplicity itself Let Z ω := {q P Λ ω(p, q)ω(q, p) = 1} then by [OPT] we have Prim C (P Λ, ω) = Ẑω. By [IW, 2] there is an action of H Λ on Prim C 0 (Λ ) C (P Λ, ω) = Λ Ẑ ω. The action is determined by a cocycle c Z 1 (H Λ, Ẑ ω ). Theorem Suppose Λ is cofinal. Then C (Λ, c) is simple iff the action of H Λ on Λ Ẑ ω is minimal. If Z ω = 0, then C (Λ, c) is simple. But this condition is not necessary as the following example shows.

10 An example In this example Λ is cofinal and ω is trivial, but C (Λ, c) is simple. Let B 2 be the 1-graph with 1 vertex and 2 edges and let T 1 = N be the 1-graph with 1 vertex and 1 edge. Let Λ := B 2 T 1 and c((µ, m), (ν, n)) := z nd(µ) where z := e 2πiθ and θ is irrational. a 1 Λ 1 a 2 Since P Λ = Z, ω is a coboundary. Thus Zω = P Λ. We have G Λ = HΛ Z and Λ = B 2 = {(x 1, x 2,... ) x i = a 1 or a 2 }. Moreover, c(x, n, y) = z n and hence C (Λ, c) is simple.

11 References [CKSS] Carlsen, Kang, Shotwell and Sims, The primitive ideals of the Cuntz - Krieger algebra of a row-finite higher-rank graph with no sources, [IW] Ionescu and Williams, Remarks on the ideal structure of Fell bundle C -algebras, [KP00] A. Kumjian and D. Pask, Higher rank graph C -algebras, [KPS] A. Kumjian, D. Pask and A. Sims, On twisted higher-rank graph C -algebras, in press. [OPT] Olesen, Pedersen and Takesaki, Ergodic actions of compact abelian groups, [R] J. Renault, A groupoid approach to C*-algebras, [SWW] Sims, Whitehead and Whittaker, Twisted C -algebras assoc. to finitely aligned higher-rank graphs, ppt.