Groupoids and higher-rank graphs

Transcription

1 Groupoids and higher-rank graphs James Rout Technion - Israel Institute of Technology 5/12/2017 (Ongoing work with Toke Meier Carlsen) James Rout Groupoids and higher-rank graphs 1 / 16

2 Higher-rank graphs Definition (Kumjian Pask) A higher-rank graph or k-graph is a countable small category Λ := (obj(λ), mor(λ), r, s) together with a functor d : Λ N k satisfying the factorisation property: for every λ Λ and m, n N k with d(λ) = m + n, there are unique elements µ, ν Λ with d(µ) = m and d(ν) = n such that λ = µν. We then write λ(0, m) := µ and λ(m, m + n) = ν. Definition (Kumjian Pask) A k-graph is called row-finite if the set vλ m := {λ Λ : d(λ) = m, r(λ) = v} is finite for all v Λ 0 and m N k. If each of the sets vλ m is nonempty, then Λ has no sources. James Rout Groupoids and higher-rank graphs 2 / 16

3 Examples of k-graphs Example (Directed graphs) If Λ is a 1-graph, then (d 1 (0), d 1 (1), r, s) is a directed graph. Conversely, if E = (E 0, E 1, r, s) is a directed graph, then E := n 0 E n, the collection of finite paths, may be viewed as small category with range and source given by r(e n... e 1 ) := r(e n ) and s(e n... e 1 ) := s(e 1 ). Taking d : E N to be the length function i.e. d(e n... e 1 ) = n, we have that (E, d) is a 1-graph. Example (Kumjian Pask) For k N, define Ω k, := {(p, q) N k N k : p q}. This is a k-graph with Ω 0 k, = Nk, r(p, q) = p, s(p, q) = q, d(p, q) = q p and composition defined by (p, q)(q, r) = (p, r). James Rout Groupoids and higher-rank graphs 3 / 16

4 Cuntz Krieger families Definition (Kumjian Pask) Let Λ be a row-finite k-graph with no sources. A Cuntz Krieger Λ-family is a collection {S λ : λ Λ} of partial isometries in a C -algebra A satisfying {S v : v Λ 0 } is a collection of mutually orthogonal projections; S λ S µ = S λµ whenever s(λ) = r(µ); S λ S λ = S s(λ) for all λ Λ; and for all v Λ 0 and n N k we have S v = λ vλ n S λs λ. James Rout Groupoids and higher-rank graphs 4 / 16

5 C -algebras associated to k-graphs Theorem (Kumjian Pask) Given a row-finite k-graph Λ with no sources, there is a C -algebra C (Λ) generated by a Cuntz Krieger Λ-family {s λ : λ Λ} which is universal in the following sense: for any Cuntz Kriger Λ-family {t λ : λ Λ} in a C -algebra A, there is a unique homomorphism π t : C (Λ) A such that π S (s λ ) = t λ for all λ Λ. This C -algebra can be written C (Λ) = span{s λ s µ : λ, µ Λ, s(λ) = s(µ)}. The diagonal subalgebra of C (Λ) is given by D(Λ) := span{s µ s µ : µ Λ}. For a k 1 -graph Λ 1 and a k 2 -graph Λ 2, we say that an isomorphism φ : C (Λ 1 ) C (Λ 2 ) is diagonal-preserving if φ(d(λ 1 )) = D(Λ 2 ). James Rout Groupoids and higher-rank graphs 5 / 16

6 Kumjian Pask families For each λ Λ \ Λ 0, we define a ghost path by a formal symbol λ. For v Λ 0, we define v := v, and extend r and s to the collection of ghost paths by setting r(λ ) := s(λ) and s(λ) := r(λ ). Composition of ghost paths is defined by λ µ := (µλ). Definition (Aranda Pino Clark an Huef Raeburn) Let Λ be a row-finite k-graph with no sources, and R a commutative ring with identity. A Kumjian Pask Λ-family {S λ, S µ : λ, µ Λ} in an R-algebra A consists of a function S : Λ {µ : µ Λ \ Λ 0 } A satisfying {S v : v Λ 0 } is a collection of mutually orthogonal idempotents; S λ S µ = S λµ and S µ S λ = S (λµ) S λ S λ = S s(λ) for all λ Λ; and whenever s(λ) = r(µ); for all v Λ 0 and n N k we have S v = λ vλ n S λs λ. James Rout Groupoids and higher-rank graphs 6 / 16

7 Kumjian Pask algebras Theorem (Aranda Pino Clark an Huef Raeburn) Given a row-finite k-graph with no sources and a commutative ring R with identity, there is an R-algebra KP(Λ) generated by a Kumjian Pask Λ-family {s λ, s µ : λ, µ Λ} which is universal in the following sense: for any Kumjian Pask Λ-family {S λ, S µ : λ, µ Λ} in an R-algebra A, there is a unique R-algebra homomorphism π S : KP(Λ) A such that π S (s λ ) = S λ and π S (s µ ) = S µ for all λ, µ Λ. This R-algebra can be written KP(Λ) = span R {s λ s µ : λ, µ Λ, s(λ) = s(µ)}. The diagonal subalgebra of KP(Λ) is given by D R (Λ) := span R {s µ s µ : µ Λ}. For a k 1 -graph Λ 1 and a k 2 -graph Λ 2, we say that a ring-isomorphism φ : KP(Λ 1 ) KP(Λ 2 ) is diagonal-preserving if φ(d ( Λ 1 )) = D(Λ 2 ). James Rout Groupoids and higher-rank graphs 7 / 16

8 The infinite path space Definition (Kumjian Pask) Let Λ be a row-finite k-graph with no sources. An infinite path is a degree-preserving functor x : Ω k, Λ. The collection of infinite paths is denoted by Λ. For v Λ 0, write vλ := {x Λ : x(0) = v}. For each p N k, define σ p : Λ Λ by σ p (x)(m, n) = x(m + p, n + p) for x Λ and (m, n) Ω k,. Lemma (Kumjian Pask) For λ Λ and x Λ with x(0) = s(λ), there exists λx Λ such that x = σ d(λ) (λx) and λ = (λx)(0, d(λ)). Define Z(λ) := {λx : x Λ, x(0) = s(λ)}. The sets Z(λ) form a basis for a locally compact Hausdorff topology on Λ. James Rout Groupoids and higher-rank graphs 8 / 16

9 The path groupoid Definition (Kumjian Pask) The path groupoid G Λ is given by G Λ := {(x, p q, y) Λ Z k Λ : p, q N k, σ p (x) = σ q (y)}, with partially-defined product (x, m, y)(y, n, z) (x, m + n, z), inverse operation (x, m, y) (y, m, x), and range and source maps r(x, m, y) := x and s(x, m, y) := y. We identify the infinite path space Λ with the unit space GΛ 0 via the map x (x, 0, x). For λ, µ Λ with s(λ) = s(µ), write Z(λ s µ) := {(λz, d(λ) d(µ), µz) : z s(λ)λ }, The sets Z(λ s µ) form a basis for a topology that makes G Λ a locally compact, Hausdorff, étale and ample groupoid. James Rout Groupoids and higher-rank graphs 9 / 16

10 Groupoid models for k-graph C -algebras and Kumjian Pask algebras For a row-finite k-graph Λ with no sources, there is an isomorphism π : C (Λ) C (G Λ ) satisfying π(s λ ) = 1 Z(λ ss(λ)) for all λ Λ. Moreover, π(d(λ)) = C 0 (G 0 Λ ) C (G Λ ), For a row-finite k-graph Λ with no sources and a commutative ring R with identity, there is an isomorphism π T : KP(Λ) A R (G Λ ) such that π T (s λ ) = 1 Z(λ ss(λ)) and π T (s λ ) = 1 Z(s(λ) sλ) for λ Λ. Moreover, π T (D R (Λ)) = A R (G 0 Λ ) A R(G Λ ). James Rout Groupoids and higher-rank graphs 10 / 16

11 Diagonal-preserving isomorphisms of k-graph C -algebras and Kumjian Pask algebras For a k-graph Λ, there is a cocycle c Λ : G Λ Z k given by c Λ (x, n, y) = n, a gauge action γ Λ : T k aut C (Λ) given by γz Λ (s λ ) = z d(λ) s λ for z T k and a Z k -grading KP(Λ) = n Z k KP(Λ) n where KP(Λ) n = span R {s µ sλ : µ, ν Λ, d(µ) d(ν) = n}. Theorem (Carlsen Ruiz Sims Tomforde, Carlsen R) Let Λ 1 and Λ 2 be k-graphs and let R be an integral domain with identity. TFAE There is an isomorphism Φ : G Λ1 G Λ2 satisfying c Λ2 Φ = c Λ1. There is a diagonal-preserving -isomorphism Ψ : C (Λ 1 ) C (Λ 2 ) satisfying γ Λ 2 z Ψ = Ψ γ Λ 1 z for z T k. There is a diagonal-preserving ring-isomorphism θ : KP(Λ 1 ) KP(Λ 2 ) satisfying θ(kp(λ 1 ) n ) = KP(Λ 2 ) n for n Z k. James Rout Groupoids and higher-rank graphs 11 / 16

12 Eventual one-sided conjugacy of k-graphs Definition (Carlsen R, Matsumoto) We say that k-graphs Λ 1 and Λ 2 are eventually one-sided conjugate if there is a homeomorphism h : Λ 1 Λ 2 and continuous maps f m : Λ 1 Nk and g m : Λ 2 Nk for each m N k satisfying σ fm(x) Λ 2 (h(σλ m 1 (x))) = σ fm(x)+m Λ 2 (h(x)) for x Λ 1 and σ gm(y) Λ 1 (h 1 (σλ m 2 (y))) = σ gm(y)+m Λ 1 (h 1 (y)) for y Λ 2. Theorem (Carlsen R) Two k-graphs Λ 1 and Λ 2 are eventually one-sided conjugate if and only if there is an isomorphism Φ : G Λ1 G Λ2 satisfying c Λ2 Φ = c Λ1. James Rout Groupoids and higher-rank graphs 12 / 16

13 Stabilised groupoids Denote by K the compact operators on l 2 (N) generated by the rank-one operators {θ i,j : i, j N} and by C the diagonal subalgebra generated by {θ i,i : i N}. For a commutative ring R with identity, we denote by M (R) the ring of finitely supported, countable infinite square matrices over R, and by D (R) the diagonal subalgebra consisting of diagonal matrices. Denote by N the full countable equivalence relation N = N N, regarded as a discrete principal groupoid with (i, j)(j, k) = (i, k), (i, j) 1 = (j, i) and unit space N. Given an ample groupoid G, the product G N is an ample groupoid under the product topology and coordinatewise operations. The unit space is identified with G 0 N. There are isomorphisms C (G N ) = C (G) K such that C 0 (G 0 N) = C 0 (G 0 ) C and A R (G N ) = A R (G) M (R) such that A R (G 0 N) = A R (G 0 ) D (R). James Rout Groupoids and higher-rank graphs 13 / 16

14 Stable isomorphisms of k-graph C -algebras and Kumjian Pask algebras For a k-graph Λ, there is a cocycle c Λ : G Λ N Z k given by c Λ (η, (i, j)) = c Λ (η) for η G Λ and (i, j) N. Corollary (Carlsen Ruiz Sims Tomforde, Carlsen R) Let Λ 1 and Λ 2 be k-graphs and let R be an integral domain with identity. TFAE There is an isomorphism Φ : G Λ1 N G Λ2 N satisfying c Λ2 Φ = c Λ1. There is a diagonal-preserving -isomorphism Ψ : C (Λ 1 ) K C (Λ 2 ) K satisfying (γ Λ 2 z Id K ) Ψ = Ψ (γ Λ 1 z Id K ) for z T k. There is a diagonal-preserving ring-isomorphism θ : KP(Λ 1 ) M (R) KP(Λ 2 ) M (R) such that θ(kp(λ 1 ) n M (R)) = KP(Λ 2 ) n M (R) for n Z k. James Rout Groupoids and higher-rank graphs 14 / 16

15 Two-sided edge shift spaces associated to k-graphs Let Ω k be the k-graph Ω k := {(m, n) Z k Z k : m n} with degree map d : Ω k N k defined by d(m, n) = n m, and range and source maps r, s defined by r(m, n) = (m, m) and s(m, n) = (n, n). Let Λ be a row-finite k-graph with finitely many vertices and no sinks or sources. We write X Λ for the space of all degree-preserving functors from Ω k to Λ. We equip X Λ with the topology generated by subsets of the form where (m, n) Ω k and λ Λ n m. Z((m, n), λ) := {x X Λ : x(m, n) = λ} James Rout Groupoids and higher-rank graphs 15 / 16

16 Two-sided conjugacy of edge shifts For m Z k, we denote by σ m : X Λ X Λ the homeomorphism given by σ m (x)(p, q) = x(p + m, q + m) for x X Λ and (p, q) Ω k. Theorem (Carlsen R) Let Λ 1 and Λ 2 be row-finite k-graphs with finitely many vertices and no sinks or sources. TFAE There is an isomorphism Φ : G Λ1 N G Λ2 N satisfying c Λ2 Φ = c Λ1. There is a homeomorphism h : X Λ1 X Λ2 such that σ m h = h σ m for all m N k. James Rout Groupoids and higher-rank graphs 16 / 16