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
Covering dimension and asymptotic dimension
1 In topology, the classical dimension theory centers around the notionof 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 �ech cohomologyHn(X) = 0 for n > dim(X).
2 In coarse geometry, Gromov introduced an analogous dimension notioncalled the asymptotic dimension. asdim: MetricSp→ Z≥0 ∪ {∞}.Some basic properties:
asdim(Zn) = asdim(Rn) = n (metric: Euclidean `2 or Manhattan `1).asdim(a tree) = 1.
Finite asymptotic dimension (FAD) has far-reaching implications. E.g.groups with FAD satis�es the Novikov conjecture (Yu).
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 2 / 14
An oversimpli�ed history of classi�cation 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-theoreticand tracial information, called the Elliott invariant.
Milestones / success stories
[AF-algebras (Elliott)], AH-algebras with slow dimension growth and of realrank zero (Elliott-Gong), TAF-algebras (Lin), Purely in�nite 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
The revolution (started by Winter): we need certain regularity properties.Winter and Zacharias developed a kind of dimension theory for (nuclear)C∗-algebras. dimnuc : CStarAlg→ Z≥0 ∪ {∞}. Some basic properties:
X topological space ⇒ dimnuc(C0(X)) = dim(X) (covering dim.).X a metric space ⇒ dimnuc(C
∗u(X)) ≤ asdim(X) (asymptotic dim.).
dimnuc(A) = 0 ⇐⇒ A is AF (= lim−→(�n.dim. C∗-alg)).A Kirchberg algebra (e.g. On) =⇒ dimnuc(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 classi�ed 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 dimnuc(A) <∞ & Gy A, when dimnuc(AoG) <∞?
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 4 / 14
dimnuc(A) <∞¾when?=⇒ dimnuc(AoG) <∞
A prominent case is when A = C(X) for metric space X and G is noncpt.
Theorem (Toms-Winter, Hirshberg-Winter-Zacharias)
If Z y X minimally and dim(X) <∞, then dimnuc(C(X) o Z) <∞.
Hirshberg-Winter-Zacharias provided a more conceptual approach byintroducing the Rokhlin dimension (more on that later).
Note: If X is in�nite, a minimal Z-action is free.Theorem (Szabó)
If Zm y X freely and dim(X) <∞, then dimnuc(C(X) o Zm) <∞.
Theorem (Szabó-W-Zacharias)
If a �nitely generated virtually nilpotent group Gy X freely anddim(X) <∞, then dimnuc(C(X) oG) <∞.
{F.g. vir.nilp. gps} Gromov= {f.g. gps with polynomial growth} 3 �nite gps,
Zm, the discrete Heisenberg group{(
1 a c0 1 b0 0 1
): a, b, c ∈ Z
}, etc.
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 5 / 14
Theorem (Szabó-W-Zacharias) repeated
F.g. vir.nilp. Gy X freely & dim(X) <∞ ⇒ dimnuc(C(X) oG) <∞.
Ingredients in the proof:1 The Rokhlin dimension dimRok(α), de�ned for a C∗-dynamical systemα : Gy A, where G is �nite (H-W-Z), Z (H-W-Z), Zm (Szabó),residually �nite (S-W-Z), compact (Hirshberg-Phillips, Gardella), R(Hirshberg-Szabó-Winter-W), ...
Theorem (Szabó-W-Zacharias)
dim+1nuc(Aoα,w G) ≤ asdim+1(�G) · dim+1
nuc(A) · dim+1Rok(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. Gαy X freely & dim(X) <∞ ⇒ dimRok(Gy 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
Parallel approaches
Similar approaches make use of other dimensions de�ned fortopological dynamical systems, e.g.,
dynamical asymptotic dimension DAD(−) (Guentner-Willett-Yu),amenability dimension dimam(−) (G-W-Y, S-W-Z, afterBartels-Lück-Reich), and(�ne) tower dimension dimtow(−) (Kerr).
They are closely related through intertwining inequalities such as:
Theorem (Szabó-W-Zacharias)
dim+1Rok(α) ≤ dim+1
am(α) ≤ dim+1Rok(α) · asdim+1(�G) .
Remarkably, the original motivations for introducing dimam and DADwere to facilitate computations of K-theory for AoG, in order toprove 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
The case of �ows
When G = R y X continuously, we also have
Theorem (Hirshberg-Szabó-Winter-W)
If R y X freely and dim(X) <∞, then dimnuc(C(X) oR) <∞.
Ingredients in the proof:1 The Rokhlin dimension dimRok(α) de�ned for any C∗-�ow α : R y A.
Theorem (H-S-W-W)
dim+1nuc(Aoα R) ≤ 2 · dim+1
nuc(A) · dim+1Rok(α) .
2 The existence of �long thin covers� on �ow spaces ,due toBartels-Lück-Reich and improved by Kasprowski-Rüping (initiallydeveloped for studying the Farrell-Jones conjecture).
Theorem (Bartels-Lück-Reich, Kasprowski-Rüping, H-S-W-W)
R y X freely and dim(X) <∞ ⇒ dimRok(Gy C(X)) <∞.
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 8 / 14
Non-free Z-actions ⇒ Problem: dimRok(α) =∞, but...
Theorem (Hirshberg-W)
Z y X loc. cpt Hausd. with dim(X) <∞ ⇒ dimnuc(C0(X) o Z) <∞.
Rough sketch of the proof: Pick a �threshold� R > 0 (to be determined).X≤R := union of (periodic) orbits of lengths ≤ R, andX>R := union of (possibly non-periodic) orbits of lengths > R.
invariant decomposition X = X≤R tX>R ⇒ Exact sequence
0→ C0(X>R) o Z→ C0(X) o Z→ C0(X≤R) o Z→ 0
Fact: dimnuc <∞ passes through extensions ⇒ Look at the two ends!1 Z y X≤R is well-behaved (in particular, X≤R/Z is Hausdor�) ⇒
dim+1nuc(C0(X≤R) o Z) ≤ 2 dim+1(X≤R) ≤ 2 dim+1(X).
Important: This bound does not depend on R!2 Fact: dimnuc <∞ is a �local approximation� property ⇒ when R is
chosen large enough (depending on the desired precision of the localapproximation), Z y X>R behaves like a free action for the purposeof the approximation ⇒ We mimic the approach for free actions.
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 9 / 14
Non-free actions of f.g. virtually nilpotent groups
theorem (Hirshberg-W)
A �nitely generated virtually nilpotent group Gy X loc. cpt Hausd. withdim(X) <∞ ⇒ dimnuc(C0(X) oG) <∞.
=⇒ Examples of groups C∗-algebras with �nite nuclear dimensions
dimnuc(C∗(Z2 oA Z)) <∞, where A =
(2 11 1
)∈ SL(2,Z). This is an
example of a group which is polycyclic but not nilpotent.
dimnuc(C∗(L)) <∞ for L = Z2 o Z = Z
⊕Z
2 oshift Z (lamplighter gp).
Both are QD but NOT strongly QD (⇒ have in�nite decomposition rank)!
Theorem (Eckhardt-McKenney without �virtually�, E-Gillaspy-M)
dimnuc(C∗(any f.g. vir.nilp. gp))
(≤ dr(C∗(any f.g. vir.nilp. gp))
)<∞.
Theorem (Eckhardt)
Decomposition rank dr(C∗(Zm oA Z)) <∞ ⇔ Zm oA Z vir.nilpotent.Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 10 / 14
For �nitely generated G, we have
virtually nilpotent GEckhardt-Gillaspy-McKenney
*2jr
True for G = ZmoAZ(Eckhardt)
��
dr(C∗(G)) <∞
��
virtually polycylic G
��elem. amenable G
with �nite Hirsch length
True for G = (abelian)o(f.g. vir.nil.)(Hirshberg-W)
*2jr
???
dimnuc(C∗(G)) <∞
Remark: dimnuc(C∗(Z o Z)) =∞. Z o Z also has in�nite 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 �niteHirsch length and groups with �nite nuclear dimension?
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 11 / 14
Non-free �ows and line foliations
Theorem (Hirshberg-W)
R y X loc. cpt Hausd. with dim(X) <∞ ⇒ dimnuc(C0(X) oR) <∞.
Application to the C∗-algebras of line foliations: A line foliation on Xconsists 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 y 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 y X.Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 12 / 14
A line foliation F on X de�nes 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∗(GF ), the groupoid C∗-algebra of the holonomy groupoid
GF associated to F .The K-theory of C∗(GF ) plays a fundamental role in the longitudinal
index theorem (Connes-Skandalis).
Proposition
If F is induced from a �ow R y X, then C∗(GF ) is a quotient ofC0(X) oR.
Theorem (Hirshberg-W)
For any orientable line foliation F on X with dim(X) <∞, we havedimnuc(C
∗(GF )) <∞.
Proof: dimnuc(C∗(GF )) ≤ dimnuc(C0(X) oR) <∞.
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 13 / 14
Thank you!
Jianchao Wu (Penn State) NC Dimensions & Topological Dynamics Lafayette, October 7 14 / 14
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