Valentin Deaconu, University of Nevada, Reno. Based on joint work with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems (2011)

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1 Based on joint work with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems (2011) Texas Tech University, Lubbock, 19 April 2012

2 Outline We define directed graphs and operator representations using projections and partial isometries on a Hilbert space. We introduce the C -algebra of a graph and illustrate with several examples. We consider the action of a group on a graph and on its C -algebra. We define the quotient graph. We define topological graphs and their C -algebras. The main result (joint work with A.Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems 2011) involves groups acting on topological graphs, crossed product C -algebras and the quotient graph.

3 Directed graphs A directed graph E is (E 0, E 1, r, s), where the sets E 0, E 1, vertices and edges are countable (finite or infinite) and r, s : E 1 E 0 are the range and source maps. We assume E locally finite.

4 Graph C -algebras A -representation of E on a Hilbert space H is given by a family of mutually orthogonal projections P v for v E 0 and a family of partial isometries S e for e E 1 such that P v = r(e)=v S e S e = P s(e) for all e E 1, S e S e for all v r(e 1 ) (v not a source). Recall that P v = P 2 v = P v and that S e S e, S e S e are projections. The operators P v and S e generate a C -algebra. The universal C -algebra is denoted C (E).

5 Properties of C (E) The projections S e S e are mutually orthogonal. e f S e S f = 0. S e S f 0 ef is a path of length 2. e f s(e) s(f ) S e S f = 0. A path µ of length m is µ = µ 1 µ 2 µ m with µ i E 1 and s(µ i ) = r(µ i+1 ) for 1 i m 1. A vertex is a path of length 0. C (E) is the closed span of where S µ = S µ1 S µ2 S µm. {S µ S ν : µ = m, ν = n, s(µ) = s(ν)},

6 Matrix algebras Consider the graph u e v We have S e S e = P u, S e S e = P v. Consider P u = on H = C 2. [ It follows that C (E) = M 2 (C). The graph with n vertices ] [ 0 0, P v = 0 1 produces the matrix algebra M n (C). ] [ 0 0, S e = 1 0 ]

7 Direct sums and inductive limits If E finite has no cycles and w 1, w 2,..., w k are sources, then C (E) = k M ni (C), i=1 where n i = P(w i ) and P(w i ) = {µ path : s(µ) = w i }. The C -algebra of compact operators K = lim M n (C) can be obtained from the infinite graph If E is infinite with no cycles, then C (E) is AF (approximately finite dimensional, an inductive limit of direct sums of matrix algebras).

8 C(T) For the graph E with one vertex v and one loop e we have S e S e = P v = S e S e = I. C (E) is the universal C -algebra generated by a unitary. We obtain C (E) = C(T), where T is the unit circle. v e

9 Toeplitz algebra T Consider H = l 2 (N) and let P v (x 0, x 1, x 2,...) = (0, x 1, x 2,...), P w (x 0, x 1, x 2,...) = (x 0, 0, 0,...), S f (x 0, x 1, x 2,...) = (0, x 0, 0,...), S e (x 0, x 1, x 2,...) = (0, 0, x 1, x 2,...). Then S e S e = P s(e) = P v, S f S f = P w and P v = S e S e + S f S f, P v + P w = I. Let V = S e + S f. Then V(x 0, x 1, x 2,...) = (0, x 0, x 1, x 2,...), V (x 0, x 1, x 2,...) = (x 1, x 2, x 3,...) and V V = I, VV = P v = I P w. For the graph below, it follows that C (E) = T, the Toeplitz algebra (generated by the unilateral shift). e v f w

10 Cuntz algebras For the graph E with one vertex v and two loops e, f we have Se S e = P v = Sf S f, P v = S e Se + S f Sf. On H = l 2 (N) = span{h n : n 0} we can take P v = I, S e (h n ) = h 2n, S f (h n ) = h 2n+1. C (E) = O 2 is generated by two isometries S e, S f with S e Se + S f Sf = I. If we have n loops, we get the Cuntz algebras O n generated by n isometries S 1, S 2,..., S n with orthogonal ranges such that n S i Si = I. i=1 e f v

11 Cuntz-Krieger algebras The Cuntz-Krieger algebra O A of a square 0 1 matrix A = (A ij ) is generated by n partial isometries S 1,..., S n with orthogonal range projections subject to S i S i = n A ij S j Sj. j=1 O A can be obtain from a graph with n vertices and incidence matrix A. These C -algebras were introduced in relation with symbolic dynamics (Markov shifts).

12 More properties of graph C -algebras Graph C -algebras generalize Cuntz-Krieger algebras. Ideals of C (E) correspond to hereditary saturated subsets in E. A subset H E 0 is hereditary if w H and if v E 0 is such that there is a path µ with s(µ) = v and r(µ) = w, then v H. H is saturated if whenever v E 0 is not a source and {s(e) : e E 1, r(e) = v} H, then v H. The K-theory of C (E) for E locally finite with no sources can be computed as K 0 (C (E)) = coker(i A E ), K 1 (C (E)) = ker(i A E ), where A E : Z E0 Z E0 is given by the vertex matrix of E.

13 Group actions Which groups act on a graph E and on C (E)? A countable group G acts on E if there is a homomorphism G Aut(E). A graph morphism is a pair of maps E 0 E 0, E 1 E 1 compatible with r, s. If G acts on E, one can define the quotient graph E/G with (E/G) 0 = E 0 /G, (E/G) 1 = E 1 /G. The action of G on E induces an action on C (E) by g P v = P g v, g S e = S g e. The action is free if G acts freely (no fixed points) on E 0.

14 Examples The group Z acts by translation on the graph with quotient the graph of C(T). A finitely generated group G acts on its Cayley graph E G and E G /G = B n, the graph with one vertex and n loops. The fundamental group π 1 (E) of a graph E acts freely on the universal covering tree T(E) and T(E)/π 1 (E) = E. For a finite graph, π 1 (E) is isomorphic to the free group F n where n = E 1 E Theorem (A. Kumjian, D. Pask). If E is locally finite and G acts freely on E, then C (E) G = C (E/G) K(l 2 (G)). The proof uses groupoids and skew products.

15 Topological graphs and their C -algebras In a topological graph E = (E 0, E 1, s, r) the spaces E 0 and E 1 are locally compact, s, r : E 1 E 0 are continuous maps, and s is a local homeomorphism. The C*-algebra C (E) is defined using the C -correspondence (Hilbert bimodule) over C 0 (E 0 ), obtained as the completion of C c (E 1 ) with inner product ξ, η (v) = ξ(e)η(e), ξ, η C c (E 1 ) and multiplications s(e)=v (ξ f )(e) = ξ(e)f (s(e)), (f ξ)(e) = f (r(e))ξ(e).

16 Examples 1. Let E 0 = E 1 = T, s(z) = z, and r(z) = e 2πiθ z for θ [0, 1] irrational. Then C (E) = A θ, the irrational rotation algebra. 2. Let E 0 = E 1 = X, for X a compact metric space, let s = id and let r = h : X X be a homeomorphism. Then C (E) = C(X) Z, the universal C -algebra generated by C(X) and a unitary u satisfying f h = u fu for f C(X). 3. Let p, q 1. Take E 0 = E 1 = T, s(z) = z p, r(z) = z q. If q / pz, then C (E) is a continuous version of the Cuntz algebras O n (it is simple and purely infinite).

17 Main result Theorem (with A. Kumjian, J.Quigg, Ergodic Theory and Dynamical Systems 2011). If G acts freely and properly on the topological graph E, then G acts properly on C (E) and the action is saturated in the sense of Rieffel. Moreover, the reduced crossed product C (E) r G and C (E/G) are strongly Morita equivalent. Two C -algebras A, B are strongly Morita equivalent if there is an A-B Hilbert bimodule X with extra properties. The proof uses the notions of generalized fixed point algebra and multiplier bimodules. Corollary. C (Ẽ) r π 1 (E) is strongly Morita equivalent to C (E), where Ẽ is the universal covering graph of E.

18 The fundamental group of a topological graph The geometric realization of a topological graph E is R(E) := E 1 [0, 1] E 0 /, where (e, 0) s(e) and (e, 1) r(e) (a kind of double mapping torus). The fundamental group π 1 (E) is π 1 (R(E)). The universal covering Ẽ of E is a simply connected graph which covers E. The group π 1 (E) acts freely on Ẽ, and the quotient graph Ẽ/G is isomorphic to E. Any subgroup H of π 1 (E) will determine an intermediate covering of E, by taking the graph Ẽ/H.

19 Examples Example 1. Let E with E 0 = E 1 = T and with s(z) = z, r(z) = e 2πiθ z for θ irrational. Then R(E) is homeomorphic to T 2, hence π 1 (E) = Z 2. The universal covering Ẽ has Ẽ 0 = Ẽ 1 = R Z, and s(y, k) = (y, k), r(y, k) = (y + θ, k + 1). Here Z 2 acts on Ẽ by (j, m) (y, k) = (y + mθ + j, k + m), and Ẽ/Z 2 = E. Example 2. Let h : X X be a homeomorphism, and let E with E 0 = E 1 = X, s = id and r = h. Then R(E) is homeomorphic to the mapping torus of h. The universal covering Ẽ has Ẽ 0 = Ẽ 1 = X Z, where X is the universal covering of X. The source and range maps are s(y, k) = (y, k), r(y, k) = ( h(y), k + 1), where h : X X is a lifting of h. We have π 1 (E) = π 1 (X) Z, and the action of π 1 (X) Z on X Z is given by (g, m) (y, k) = (g h m (y), k + m).

20 Examples Example 3. Let again E 0 = E 1 = T with s(z) = z p, r(z) = z q for p, q positive integers. Then R(E) is obtained from a cylinder, where the two boundary circles are identified using the maps s and r. Figure: The case p = 2, q = 3.

21 Examples Then π 1 (E) is isomorphic to the Baumslag-Solitar group B(p, q) = a, b ab p a 1 = b q. For p = 1 or q = 1, this group is a semi-direct product and it is amenable. For p 1, q 1 and (p, q) = 1, it is not amenable. The universal covering space of R(E) is obtained from the Cayley graph of B(p, q) by filling out the squares. It is the cartesian product T R, where T is the Bass-Serre tree of B(p, q), viewed as an HNN-extension of π 1 (T). Recall that B(p, q) is the quotient of the free product π 1 (T) Z by the relation as (b)a 1 = r (b), where a is the generator of Z, b π 1 (T), and s, r : π 1 (T) π 1 (T).

22 Examples a ab ab 2 =b 3 a ba bab bab 2 =b 4 a b 2 a b 2 ab b 2 ab 2 =b 5 a e b b 2 b 3 b 4 b 5 ba -1 ba -1 b b 2 a -1 b 2 a -1 b b 2 a -1 b 2 b 2 a -1 b 3 ba -1 b 2 ba -1 b 3 a ab ab 2 a 2 a 2 b a 2 b 2 Figure: Cayley a complex for B(2, 3). e b b 2 b 3 a ab ab 2 ab 3 a ab ab 2 ba bab bab 2 b

23 Examples The 1-skeleton is the directed Cayley graph of B(2, 3), where the generators a, b multiply on the right. The group action is given by left multiplication. In the corresponding tree T, each vertex has 5 edges. The vertex set T 0 is identified with the left cosets g b B(2, 3)/ b, and the edge set T 1 with the left cosets g b 2 B(2, 3)/ b 2. The source and range maps are given by s(g b 2 ) = g b, r(g b 2 ) = ga 1 b for g B(2, 3). We have Ẽ 0 = T 0 R, Ẽ 1 = T 1 R with s(t, y) = (s(t), py), r(t, y) = (r(t), qy) for t T 1 and y R. The group B(p, q) acts freely and properly on Ẽ, and the quotient graph Ẽ/B(p, q) is isomorphic to E. In particular, Ẽ is not a skew product. We have C (Ẽ) r B(p, q) strongly Morita equivalent to C (E).

24 References C. Anantharaman-Delaroche, J. Renault, Amenable groupoids. G. Baumslag, Topics in combinatorial group theory. M. Bridson, A. Haefliger, Metric spaces of non-positive curvature. T. Katsura, A class of C -algebras generalizing both graph algebras and homeomorphism C -algebras I, Fundamental results, Trans. Amer. Math. Soc. 356 (2004), no. 11, T. Katsura, A class of C -algebras generalizing both graph algebras and homeomorphism C -algebras IV, pure infiniteness, J. Funct. Anal. 254 (2008), no. 5,

25 References cont d T. Katsura, Continuous graphs and crossed products of Cuntz algebras, Recent aspects of C -algebras (Japanese) (Kyoto, 2002). Sūrikaisekikenkyūsho Kōkyūroku No (2002), A. Kishimoto, A. Kumjian, Crossed products of Cuntz algebras by quasi-free automorphisms Operator algebras and their applications (Waterloo, ON, 1994/1995), A. Kumjian, D. Pask, C -algebras of directed graphs and group actions. Ergodic Theory Dynam. Systems 19 (1999), no. 6, R. C. Lyndon, P. E. Schupp, Combinatorial group theory. M. Pimsner, A Class of C*-Algebras Generalizing both Cuntz-Krieger Algebras and Crossed Products by Z.