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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