Twisted Higher Rank Graph C*-algebras

Transcription

1 Alex Kumjian 1, David Pask 2, Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong East China Normal University, Shanghai, 21 May 2012

2 Introduction. Preliminaries Introduction k-graphs Remarks We define the C*-algebra C ϕ(λ) of a higher rank graph Λ twisted by a 2-cocycle ϕ which takes values in T and derive some basic properties. Examples of this construction include all noncommutative tori, crossed products of Cuntz algebras by quasifree automorphisms and Heegaard quantum 3-spheres (see [BHMS]). We also discuss the cohomology theory, where the twisting cocycle ϕ resides, and the homology theory on which it is based. Our definition of the homology of a k-graph Λ is modeled on the cubical singular homology of a topological space (see [Mas91, VII.2]). It agrees with the homology of the associated cubical set (see [Gr05]). This talk is based on joint work with David Pask and Aidan Sims of the University of Wollongong. Many of the the results discussed here were obtained while I was also employed there.

3 Introduction k-graphs Remarks k-graphs. 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 there exist unique µ, ν Λ satisfying: d(µ) = m and d(ν) = n, λ = µν. d(λ) = m + 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.

4 Introduction k-graphs Remarks 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. Commuting squares form an essential piece of structure for k 2. Let C m denote the directed cycle with m vertices viewed as a 1-graph. Example of a 2-graph Λ: Only the edges, Λ e1 and Λ e2, are shown. f a u e v b Note that Λ = C 2 C 1.

5 Introduction k-graphs Remarks More examples The k-graph T k := N k is regarded as the k-graph analog of a torus. Here is a simple k-graph with an infinite number of vertices: k := {(m, n) Z k Z k m n} with structure maps s(m, n) = n r(m, n) = m d(m, n) = n m (l, n) = (l, m)(m, n). This may be regarded as the k-graph analog of Euclidean space.

6 Homology Basic Results Cohomology Cubes and Faces. Let Λ be a k-graph. For 0 n k an element λ Λ with d(λ) = e i1 + + e in where i 1 < < i n is called an n-cube. Let Q n (Λ) denote the set of n-cubes. Note that 0-cubes are vertices and 1-cubes are edges. For n < 0 or n > k, we have Q n (Λ) =. Let λ Q n (Λ). We define the faces F 0 j (λ), F1 j (λ) Q n 1 (Λ), where 1 j n, to be the unique elements such that where d(λ l ) = e ij for l = 0, 1. λ = F 0 j (λ)λ 0 = λ 1 F 1 j (λ) Fact: If i < j, then F l i F m j = F m j 1 Fl i.

7 Homology complex. Preliminaries Homology Basic Results Cohomology For 1 n k define n : ZQ n (Λ) ZQ n 1 (Λ) such that for λ Q n (Λ) n (λ) = n 1 ( 1) j+l Fj l (λ). j=1 l=0 It is straightforward to show that n 1 n = 0. Hence, (ZQ (Λ), ) is a complex and we define the homology of Λ by H n (Λ) = ker n / Im n+1. The assignment Λ H (Λ) is a covariant functor. Example: Recall that C m is a cycle with m vertices. One may check that { H n (C m ) Z if n = 0, 1 = 0 otherwise.

8 Homology Basic Results Cohomology The Künneth Theorem. Using basic homological algebra one may prove: Theorem (Künneth Formula) Let Λ i be a k i -graph for i = 1, 2. For n 0 there is an exact sequence: 0 m 1+m 2=n H m1 (Λ 1 ) H m2 (Λ 2 ) α β H n (Λ 1 Λ 2 ) m 1+m 2=n 1 Tor(H m1 (Λ 1 ), H m2 (Λ 2 )) 0. Let Λ be the 2-graph example above and recall that Λ = C 2 C 1. By the Künneth Theorem we have H 0 (Λ) = Z, H 1 (Λ) = Z 2, H 2 (Λ) = Z.

9 Homology Basic Results Cohomology Acyclic k-graphs and free actions. A k-graph Λ is said to be acyclic if H 0 (Λ) = Z and H n (Λ) = 0 for n > 0. It is easy to show that k is acyclic. Theorem Let Λ be an acyclic k-graph and suppose that there is a free action of the group G on Λ. Then for each n 0 there is an isomorphism: H n (Λ/G) = H n (G). Example. Take Λ = k and let G = Z k act on k by translation. We have k /Z k = T k and so H n (T k ) = H n (Z k ) = Z (k n). Of course, this also follows by the Künneth Theorem.

10 Homology Basic Results Cohomology Cohomology. Let Λ be a k-graph and let A be an abelian group. For n N set C n (Λ, A) = Hom(ZQ n (Λ), A) and define δ n : C n (Λ, A) C n+1 (Λ, A) by δ n (ϕ) = ϕ n+1. It is straightforward to show that (C (Λ, A), δ ) is a complex. We define the cohomology of Λ by H n (Λ, A) := Z n (Λ, A)/B n (Λ, A), where Z n (Λ, A) := ker δ n and B n (Λ, A) := Im δ n 1. Note Λ H (Λ, A) is a contravariant functor (it is covariant in A).

11 Homology Basic Results Cohomology The UCT and a long exact sequence. Theorem (Universal Coefficient Theorem) Let Λ be a k-graph and let A be an abelian group. Then for n 0, there is a short exact sequence 0 Ext(H n 1 (Λ), A) H n (Λ, A) Hom(H n (Λ), A) 0. By a standard argument, a short exact sequence of coefficient groups gives rise to a long exact sequence 0 A B C 0 0 H 0 (Λ, A) H 0 (Λ, B) H 0 (Λ, C) H 1 (Λ, A) H n 1 (Λ, C) H n (Λ, A) H n (Λ, B) H n (Λ, C)

12 The C -algebra C ϕ(λ). Preliminaries Definition Main Results Examples Suppose that Λ satisfies ( ): For all v Λ 0, n N k, vλ n is finite and nonempty where vλ n := r 1 (v) Λ n. Definition Let ϕ Z 2 (Λ, T). Define C ϕ(λ) to be the universal C -algebra generated by a family of operators {t λ : λ Λ ei, 1 i k} and a family of orthogonal projections {p v : v Λ 0 } satisfying: 1 For λ Λ ei, t λ t λ = p s(λ). 2 Suppose µν = ν µ where d(µ) = d(µ ) = e i, d(ν) = d(ν ) = e j and i < j. Then t ν t µ = ϕ(µν)t µ t ν. 3 For v Λ 0 and i = 1,..., k, p v = λ vλ e i t λ t λ.

13 Definition Main Results Examples Main Results. Fact: The isomorphism class of C ϕ(λ) only depends on [ϕ] H 2 (Λ, T). There is a gauge action γ of T k on C ϕ(λ): For all z T k γ z (p v ) = p v for all v Λ 0, γ z (t λ ) = z i t λ for all λ Λ ei, i = 1,..., k. Moreover, the fixed point algebra C ϕ(λ) γ is AF (cf. [KP00]). Theorem (Gauge Invariant Uniqueness Theorem) Let π : C ϕ(λ) B be an equivariant -homomorphism. Then π is injective iff π(p v ) 0 for all v Λ 0. Theorem There is a T-valued groupoid 2-cocycle σ ϕ on G Λ such that C ϕ(λ) = C (G Λ, σ ϕ ).

14 Definition Main Results Examples Rotation algebras Recall that T k = N k. There is precisely one 2-cube in T 2, namely (1, 1). Fix θ [0, 1). Let ϕ Z 2 (T 2, T) be given by ϕ(1, 1) = e 2πiθ. Then C ϕ(t 2 ) is the universal C -algebra generated by unitaries t e1 and t e2 satisfying t e2 t e1 = e 2πiθ t e1 t e2. That is, C ϕ(t 2 ) is the rotation algebra A θ. When θ = 0, C ϕ(t 2 ) = C(T 2 ). When θ is irrational, C ϕ(t 2 ) is the well-known irrational rotation algebra. More generally, every noncommutative torus arises as a twisted k-graph C -algebra C ϕ(t k ).

15 Definition Main Results Examples Crossed products of Cuntz algebras Let Λ = B 2 C 1 where B 2 is the 1-graph with one vertex and two edges. Note that C (B 2 ) = O 2 and so C (Λ) = O 2 C(T). a 1 v b There are two 2-cubes in Λ, a j b for j = 1, 2. The boundary maps are trivial; so we have Z 2 (Λ, T) = H 2 (Λ, T) = T 2 where a 2 Λ Z 2 (Λ, T) ϕ (ϕ(a 1 b), ϕ(a 2 b)) Fix ϕ Z 2 (Λ, T), say ϕ(a j b) = z j. C ϕ(λ) is isomorphic to the universal C -algebra generated by two isometries, s 1, s 2, and a unitary u such that s 1 s 1 + s 2 s 2 = 1 and us j = z j s j u. So C ϕ(λ) = O 2 α Z where α(s j ) = z j S j. Hence, every crossed product of O 2 by a quasifree automorphism is isomorphic to one of the form C ϕ(λ).

16 Heegaard quantum 3-spheres Definition Main Results Examples The quantum 3-sphere Spqθ 3 where p, q, θ [0, 1) is defined in [BHMS]. The authors prove that Spqθ 3 = S00θ 3. Note S00θ 3 is the universal C -algebra generated by S and T satisfying (1 SS )(1 TT ) = 0, ST = e 2πiθ TS, S S = T T = 1, ST = e 2πiθ T S. It was known that S000 3 is isomorphic to C (Λ) where Λ is the 2-graph a b w h u c v g f But what about S 3 00θ?

17 Definition Main Results Examples Quantum spheres are twisted 2-graph C -algebras The degree map gives a homomorphism f : Λ T 2 and the induced map is an isomorphism. f : H 2 (T 2, T) H 2 (Λ, T). There are three 2-cubes α = ah = hb, β = cg = fc and τ = af = fa. Fix θ [0, 1). The 2-cocycle on T 2 determined by (1, 1) e 2πiθ pulls back to a 2-cocycle ϕ on Λ satisfying ϕ(α) = ϕ(β) = ϕ(τ) = e 2πiθ. Let {t λ : λ Λ ei, 1 i k} and {p v : v Λ 0 } be the generators of C ϕ(λ). By the universal property there is a unique map Ψ : S 3 00θ C ϕ(λ) such that Ψ(S) = t a + t b + t c and Ψ(T) = t f + t g + t g. Moreover, Ψ is an isomorphism.

18 Another Cohomology Realization Finis Categorical cocycle cohomology. The categorical cocycle cohomology, H cc(λ, A), is just the usual cocycle cohomology for groupoids (see [Ren80]) extended to small categories. We have proven that for n = 0, 1, 2 H n (Λ, A) = H n cc(λ, A). A map c : Λ Λ A is a categorical 2-cocycle if 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 categorical 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 cc(λ, A) is the quotient group (2-cocycles modulo 2-coboundaries).

19 Another Cohomology Realization Finis The C*-algebra C (Λ, c). Suppose Λ satisfies ( ) and let c be a T-valued categorical 2-cocycle. Definition (see [KPS]) 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 s v = λ vλ n s λ s λ If [ϕ] is mapped to [c] in the identification H 2 (Λ, A) = H 2 cc(λ, A), then C ϕ(λ) = C (Λ, c)..

20 Another Cohomology Realization Finis Topological realizations. One may construct the topological realization X Λ of a k-graph Λ (see [KKQS]) by analogy with the geometric realization of a simplicial set. Let I = [0, 1]. For i = 1,..., n and l = 0, 1 define ε l i : I n 1 I n by ε l i (x 1,..., x n 1 ) = (x 1,..., x i 1, l, x i,..., x n 1 ). Then the topological realization is the quotient of k Q n (Λ) I n n=0 by the equivalence relation generated by (λ, ε l i (x)) (F l i (λ), x) where λ Q n (Λ) and x I n 1. We prove that there is a natural isomorphism H n (Λ) = H n (X Λ ).

21 References. Preliminaries Another Cohomology Realization Finis [BHMS] P. F. Baum, P. M. Hajac, R. Matthes and W. Szymański, The K-theory of Heegaard-type quantum 3-spheres, [Gr05] M. Grandis, Directed combinatorial homology and noncommutative tori, [KKQS] S. Kaliszewski, A. Kumjian, J. Quigg and A. Sims, Topological realizations of higher-rank graphs, in preparation. [KP00] A. Kumjian and D. Pask, Higher rank graph C -algebras, [KPS] A. Kumjian, D. Pask and A. Sims, Homology and cohomology of higher-rank graphs, preprint. [Mas91] W. Massey, A Basic Course in Algebraic Topology, [Ren80] J. Renault, A groupoid approach to C*-algebras, 1980.

22 Another Cohomology Realization Finis Thanks!