Noncommutative Dimensions and Topological Dynamics

Transcription

1 Noncommutative Dimensions and Topological Dynamics Jianchao Wu Penn State University ECOAS, University of Louisiana at Lafayette, October 7, 2017 Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 1 / 14

2 Covering dimension and asymptotic dimension 1 In topology, the classical dimension theory centers around the notion of covering dimension for topological spaces. dim : TopSp Z 0 { }. Some basic properties: X is a manifold or a CW-complex dim(x) is the usual dimension. dim(x) = 0 X is totally disconnected (e.g., discrete Cantor) Finite covering dimension is often a regularity property in (algebraic) topology, because, for example, it implies the ƒech cohomology Ȟ n (X) = 0 for n > dim(x). 2 In coarse geometry, Gromov introduced an analogous dimension notion called the asymptotic dimension. asdim: MetricSp Z 0 { }. Some basic properties: asdim(z n ) = asdim(r n ) = n (metric: Euclidean l 2 or Manhattan l 1 ). asdim(a tree) = 1. Finite asymptotic dimension (FAD) has far-reaching implications. E.g. groups with FAD satises the Novikov conjecture (Yu). Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 2 / 14

3 An oversimplied history of classication of C -algebras Goal/fantasy: classify all C -algebras up to -isomorphisms. The Elliott Program Classify all nuclear simple separable (unital) C -algebras using K-theoretic and tracial information, called the Elliott invariant. Milestones / success stories [AF-algebras (Elliott)], AH-algebras with slow dimension growth and of real rank zero (Elliott-Gong), TAF-algebras (Lin), Purely innite algebras (Kirchberg-Phillips), etc... The crisis Jiang-Su: an -dim nuclear unital simple separable algebra Z, s.t. A Z has the same Elliott invariant as A. = need to restrict attention to (tensorially) Z-stable algebras: A = A Z (note: Z = Z Z). Without this condition, Villadsen and Toms found counterexamples. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 3 / 14

4 The revolution (started by Winter): we need certain regularity properties. Winter and Zacharias developed a kind of dimension theory for (nuclear) C -algebras. dim nuc : CStarAlg Z 0 { }. Some basic properties: X topological space dim nuc (C 0 (X)) = dim(x) (covering dim.). X a metric space dim nuc (Cu(X)) asdim(x) (asymptotic dim.). dim nuc (A) = 0 A is AF (= lim(n.dim. C -alg)). A Kirchberg algebra (e.g. O n ) = dim nuc (A) = 1. Finite nuclear dimension is preserved under taking:,, quotients, hereditary subalgebras, direct limits, extensions, etc. Theorem (Gong-Lin-Niu, Elliott-Gong-Lin-Niu, Tikuisis-White-Winter,..., Kirchberg-Phillips,... ) The class of simple separable unital C -algebras with nite nuclear dimension (FAD) and satisfying UCT is classied by the Elliott invariant. Crossed products are a major source of interesting C -algebras. We ask: Question: When does FND pass through taking crossed products? More precisely, if dim nuc (A) < & G A, when dim nuc (A G) <? Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 4 / 14

5 dim nuc (A) < ¾when? = dim nuc (A G) < A prominent case is when A = C(X) for metric space X and G is noncpt. Theorem (Toms-Winter, Hirshberg-Winter-Zacharias) If Z X minimally and dim(x) <, then dim nuc (C(X) Z) <. Hirshberg-Winter-Zacharias provided a more conceptual approach by introducing the Rokhlin dimension (more on that later). Note: If X is innite, a minimal Z-action is free. Theorem (Szabó) If Z m X freely and dim(x) <, then dim nuc (C(X) Z m ) <. Theorem (Szabó-W-Zacharias) If a nitely generated virtually nilpotent group G X freely and dim(x) <, then dim nuc (C(X) G) <. {F.g. vir.nilp. gps} Gromov = {f.g. gps with polynomial ) growth} } nite gps, Z m, the discrete Heisenberg group : a, b, c Z, etc. {( 1 a c 0 1 b Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 5 / 14

6 Theorem (Szabó-W-Zacharias) repeated F.g. vir.nilp. G X freely & dim(x) < dim nuc (C(X) G) <. Ingredients in the proof: 1 The Rokhlin dimension dim Rok (α), dened for a C -dynamical system α: G A, where G is nite (H-W-Z), Z (H-W-Z), Z m (Szabó), residually nite (S-W-Z), compact (Hirshberg-Phillips, Gardella), R (Hirshberg-Szabó-Winter-W),... Theorem (Szabó-W-Zacharias) dim +1 nuc(a α,w G) asdim +1 ( G) dim +1 nuc(a) dim +1 Rok (α). 2 The marker property (and the topological small boundary property), studied by Lindenstrauss, Gutman, Szabó, and others. Theorem (Szabó-W-Zacharias) F.g. vir.nilp. G α X freely & dim(x) < dim Rok (G C(X)) <. 3 Bound asdim +1 ( G) for f.g. vir.nilp G (S-W-Z, Delabie-Tointon). Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 6 / 14

7 Parallel approaches Similar approaches make use of other dimensions dened for topological dynamical systems, e.g., dynamical asymptotic dimension DAD( ) (Guentner-Willett-Yu), amenability dimension dim am ( ) (G-W-Y, S-W-Z, after Bartels-Lück-Reich), and (ne) tower dimension dim tow ( ) (Kerr). They are closely related through intertwining inequalities such as: Theorem (Szabó-W-Zacharias) dim +1 Rok (α) dim+1 am(α) dim +1 Rok (α) asdim+1 ( G). Remarkably, the original motivations for introducing dim am and DAD were to facilitate computations of K-theory for A G, in order to prove K-theoretic isomorphism conjectures (the Baum-Connes conjecture and the Farrell-Jones conjecture). Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 7 / 14

8 The case of ows When G = R X continuously, we also have Theorem (Hirshberg-Szabó-Winter-W) If R X freely and dim(x) <, then dim nuc (C(X) R) <. Ingredients in the proof: 1 The Rokhlin dimension dim Rok (α) dened for any C -ow α: R A. Theorem (H-S-W-W) dim +1 nuc(a α R) 2 dim +1 nuc(a) dim +1 Rok (α). 2 The existence of long thin covers on ow spaces,due to Bartels-Lück-Reich and improved by Kasprowski-Rüping (initially developed for studying the Farrell-Jones conjecture). Theorem (Bartels-Lück-Reich, Kasprowski-Rüping, H-S-W-W) R X freely and dim(x) < dim Rok (G C(X)) <. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 8 / 14

9 Non-free Z-actions Problem: dim Rok (α) =, but... Theorem (Hirshberg-W) Z X loc. cpt Hausd. with dim(x) < dim nuc (C 0 (X) Z) <. Rough sketch of the proof: Pick a threshold R > 0 (to be determined). X R := union of (periodic) orbits of lengths R, and X >R := union of (possibly non-periodic) orbits of lengths > R. invariant decomposition X = X R X >R Exact sequence 0 C 0 (X >R ) Z C 0 (X) Z C 0 (X R ) Z 0 Fact: dim nuc < passes through extensions Look at the two ends! 1 Z X R is well-behaved (in particular, X R /Z is Hausdor) dim +1 nuc(c 0 (X R ) Z) 2 dim +1 (X R ) 2 dim +1 (X). Important: This bound does not depend on R! 2 Fact: dim nuc < is a local approximation property when R is chosen large enough (depending on the desired precision of the local approximation), Z X >R behaves like a free action for the purpose of the approximation We mimic the approach for free actions. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 9 / 14

10 Non-free actions of f.g. virtually nilpotent groups theorem (Hirshberg-W) A nitely generated virtually nilpotent group G X loc. cpt Hausd. with dim(x) < dim nuc (C 0 (X) G) <. = Examples of groups C -algebras with nite nuclear dimensions dim nuc (C (Z 2 A Z)) <, where A = ( ) SL(2, Z). This is an example of a group which is polycyclic but not nilpotent. dim nuc (C (L)) < for L = Z 2 Z = Z2 shift Z (lamplighter gp). Both are QD but NOT strongly QD ( have innite decomposition rank)! Theorem (Eckhardt-McKenney without virtually, E-Gillaspy-M) ( ) dim nuc (C (any f.g. vir.nilp. gp)) dr(c (any f.g. vir.nilp. gp)) <. Z Theorem (Eckhardt) Decomposition rank dr(c (Z m A Z)) < Z m A Z vir.nilpotent. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 10 / 14

11 For nitely generated G, we have Eckhardt-Gillaspy-McKenney virtually nilpotent G virtually polycylic G True for G = Z m A Z (Eckhardt) dr(c (G)) < elem. amenable G with nite Hirsch length True for G = (abelian) (f.g. vir.nil.) (Hirshberg-W)??? dim nuc (C (G)) < Remark: dim nuc (C (Z Z)) =. Z Z also has innite Hirsch length. Question (Eckhardt-Gillaspy-McKenney) For f.g. group G, dr(c (G)) < G is virtually nilpotent? Question What is the relation between elementary amenable groups with nite Hirsch length and groups with nite nuclear dimension? Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 11 / 14

12 Non-free ows and line foliations Theorem (Hirshberg-W) R X loc. cpt Hausd. with dim(x) < dim nuc (C 0 (X) R) <. Application to the C -algebras of line foliations: A line foliation on X consists of an atlas of compatible charts of the form (0, 1) U. (Figures taken from Groupoids, Inverse Semigroups, and their Operator Algebras by Alan Paterson) A ow R X without xed points an orientable line foliation. Orientation for a line foliation = global choice of directions for all lines. Theorem (Whitney) Every orientable line foliation is induced by a ow R X. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 12 / 14

13 A line foliation F on X denes an equivalence relation F on X of being on the same leaf. X/ F is typically pathological. Connes: consider the noncommutative quotient; more precisely, consider C (G F ), the groupoid C -algebra of the holonomy groupoid G F associated to F. The K-theory of C (G F ) plays a fundamental role in the longitudinal index theorem (Connes-Skandalis). Proposition If F is induced from a ow R X, then C (G F ) is a quotient of C 0 (X) R. Theorem (Hirshberg-W) For any orientable line foliation F on X with dim(x) <, we have dim nuc (C (G F )) <. Proof: dim nuc (C (G F )) dim nuc (C 0 (X) R) <. Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 13 / 14

14 Thank you! Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 14 / 14