The fundamental group of topological graphs and C -algebras

Transcription

1 The fundamental group of topological graphs and C -algebras Based on joint work with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical Systems (2011) University of Texas at San Antonio, 27 April 2012

2 Outline We define the concept of topological graph. We introduce the fundamental group π 1 (E) and the universal covering Ẽ of a topological graph E using a geometric realization R(E). We give some examples of fundamental groups, one being the Baumslag-Solitar group. We define the C -algebra of a topological graph, with examples. We define the action of a group G on a topological graph E, the quotient graph E/G and the action of G on C (E). The main result (based on joint paper with A. Kumjian and J. Quigg, Ergodic Theory and Dynamical systems 2011): C (Ẽ) r π 1 (E) is strongly Morita equivalent to C (E).

3 Topological graphs Topological graphs generalize finite directed graphs E = (E 0, E 1, r, s), where E 0 is the set of vertices, E 1 the set of edges and r, s : E 1 E 0 are the range and source maps. In a topological graph E = (E 0, E 1, r, s) the spaces E 0 and E 1 are locally compact, r, s : E 1 E 0 are continuous maps, and s is a local homeomorphism. Example 1. Let E 0 = E 1 = T, where T is the unit circle, with s(z) = z, r(z) = e 2πiθ z for θ irrational. Example 2. Let X be compact, let h : X X be a homeomorphism, and let E 0 = E 1 = X, s = id and r = h. Example 3. Let again E 0 = E 1 = T with s(z) = z n, r(z) = z m for m, n 1.

4 The fundamental group Given a path connected space X, the fundamental group π 1 (X) is the set of homotopy classes of loops in X based at a fixed point. For example, π 1 (T) = Z. If B n is a bouquet of n circles, then π 1 (B n ) = F n, the free group with n generators. The geometric realization of a topological graph E is the space 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 by definition π 1 (R(E)).

5 Coverings A covering of X is a space Y together with a map φ : Y X with the property that each point in X has an open neighborhood U such that φ 1 (U) is a disjoint union of homeomorphic copies of U. For example, φ : T T, φ(z) = z n is a covering of T; ψ : R T, ψ(t) = e 2πit is another covering. A graph morphism φ : E F is a pair of continuous maps φ = (φ 0, φ 1 ) such that the following diagram is commutative. E 0 E 1 φ 0 φ 1 F 0 r r F 1 s E 0 φ 0 s F 0 A graph morphism φ is a covering if both φ 0, φ 1 are covering maps. The universal covering Ẽ of E is a simply connected graph (π 1 (Ẽ) = 1) which covers E. Let E be such that R = R(E) has a universal covering space R. Then R is homeomorphic to the geometric realization of a simply connected graph Ẽ, which covers E.

6 Examples Example 1. If E is a finite graph, then R(E) is a drawing of E (as a 1-complex) and π 1 (E) is free with E 1 E generators. The universal covering Ẽ is a tree. Example 2. 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 the torus 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).

7 Examples Example 3. Let X be compact with universal cover X, 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 M h of h. The universal covering Ẽ has Ẽ 0 = Ẽ 1 = X Z with 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) h Z, the semidirect product. The proof uses the exact sequence of homotopy groups of the fibration X M h T, which splits.

8 Examples Example 4. Let again E 0 = E 1 = T with s(z) = z n, r(z) = z m for m, n 1. Then R(E) is obtained from a cylinder, where the two boundary circles are identified using the maps s and r. Figure: The case n = 2, m = 3.

9 Examples The fundamental group π 1 (E) is isomorphic to the Baumslag-Solitar group B(n, m) = a, b ab n a 1 = b m. For n = 1 or m = 1, this group is a semidirect product and it is amenable. For m, n 2 and gcd(m, n) = 1, it is not amenable. The universal covering space of R(E) is obtained from the Cayley graph of B(n, m) by filling out the squares. It is T R, where T is the Bass-Serre tree of B(n, m), viewed as an HNN-extension. Let G = b = Z, let H = b n = nz and let τ : H G, τ(b n ) = b m. The HNN extension G H τ is G, a : aha 1 = τ(h), h H.

10 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

11 Examples The 1-skeleton is the directed Cayley graph of B(2, 3), where the generators a, b multiply on the right. 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 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), 2y), r(t, y) = (r(t), 3y) for t T 1 and y R.

12 The C -algebra of a topological graph For a finite graph E, its C -algebra C (E) was initially defined using projections P v for v E 0 and partial isometries S e for e E 1 satisfying Se S e = P s(e) for all e E 1, P v = S e Se for all v r(e 1 ). r(e)=v The C*-algebra C (E) of a topological graph E is defined via the Pimsner construction using the C -correspondence H = H E over C 0 (E 0 ), obtained as a completion of C c (E 1 ) with the 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).

13 Examples C (E) is a quotient of the Toeplitz algebra generated by the creation operators T ξ (η) = ξ η on the Fock space F H = H m, where m=0 H 0 = C 0 (E 0 ) and the tensor products are balanced over C 0 (E 0 ). Example 1. Let E have one vertex and n 2 edges. Then C 0 (E 0 ) = C and H = C n with the usual Hilbert space structure. Let h 1, h 2,..., h n be the canonical basis in H = C n, and consider T i : F H F H, T i (h) = h i h for h H m and i = 1,..., n. n Then Ti T j = δ ij I and T i Ti = I P, where P is the projection on H 0 = C. i=1 The operators T i generate the Toeplitz algebra T n and the quotient by K(F H ) is the Cuntz algebra O n generated by n isometries S 1,..., S n n with orthogonal ranges such that S i Si = I. i=1

14 Examples Example 2. If E 0, E 1 are countable (finite or infinite), then C (E) is the usual graph algebra generalizing Cuntz-Krieger algebras O A for A the incidence matrix. Recall that O A 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 Example 3. 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. A θ is generated by two unitaries U, V such that UV = e 2πiθ VU.

15 Examples Example 4. 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 0 (X). Example 5. Let m, n 2 and take E 0 = E 1 = T, s(z) = z n, r(z) = z m. If gcd(m, n) = 1, then C (E) is a continuous version of the Cuntz algebras. It is simple and purely infinite with K-theory K 0 = Z/(n 1)Z, K 1 = Z/(m 1)Z. It is generated by a unitary U and an isometry S such that n 1 U k SS U k = I, U n S = SU m. k=0

16 Group actions A locally compact group G acts on a topological graph E if there is a continuous homomorphism G Aut(E). The action is called free if there are no fixed points. The action is called proper if the maps G E 0 E 0 E 0, (g, v) (g v, v) and G E 1 E 1 E 1, (g, e) (g e, e) are proper. If G acts freely and properly on E, then there is a quotient graph E/G with vertex space E 0 /G and edge space E 1 /G. For example, π 1 (E) acts freely and properly on Ẽ and Ẽ/G = E. Any subgroup H of π 1 (E) acts freely and properly on Ẽ and will determine a covering φ : Ẽ/H E.

17 The main result The action of G on E induces an action of G on H E and on C (E) by g ξ(e) = ξ(g 1 e) for ξ C c (E 1 ). Using Rieffel s notion of proper saturated action and generalized fixed point algebra, we proved Theorem If a locally compact group G acts freely and properly on a topological graph E, then C (E) r G is strongly Morita equivalent with C (E/G). Two C -algebras A, B are strongly Morita equivalent if there is an A-B Hilbert bimodule X with extra properties. Corollary C (Ẽ) r π 1 (E) is strongly Morita equivalent to C (E).

18 Proper actions The action α of a locally compact group G on a C -algebra A is proper if there is a dense α-invariant -subalgebra A 0 of A such that for every a, b A 0 the functions are integrable on G, and x aα x (b) and x (x) 1/2 aα x (b) for all a, b A 0 there exists a, b r M(A 0 ), where M(A 0 ) := {m M(A) : a A 0 ma A 0 } such that c a, b r = cα x (a b)dx for all c A 0. G

19 Proper action cont d For such an action, A α := span{ a, b r : a, b A 0 } M(A) is called the generalized fixed-point algebra. Define a (left) inner product on A 0 with values in A α,r G by l a, b (x) = (x) 1/2 aα x (b ). The set I := span{ l a, b : a, b A 0 } is an ideal in A α,r G, and the closure Z of A 0 in the norm a 2 := a, a r is an I A α imprimitivity bimodule. The action is called saturated if I = A α,r G.

20 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,

21 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.