Crossed Products and Noncommutative Dimensions

Jianchao Wu(joint with Ilan Hirshberg)

Penn State University

Fields Institute, Toronto, August 4, 2017

Jianchao Wu (Penn State) Crossed Products and NC Dimensions Toronto, August 4 1 / 12

Nuclear dimension and the Elliott classi�cation program

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 unital simple separable 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 FAD pass through taking crossed products?

More precisely, if dimnuc(A) <∞ & Gy A, when dimnuc(AoG) <∞?Jianchao Wu (Penn State) Crossed Products and NC Dimensions Toronto, August 4 2 / 12

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) Crossed Products and NC Dimensions Toronto, August 4 3 / 12

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) Crossed Products and NC Dimensions Toronto, August 4 4 / 12

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) Crossed Products and NC Dimensions Toronto, August 4 5 / 12

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.

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) Crossed Products and NC Dimensions Toronto, August 4 6 / 12

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) <∞.

=⇒ 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) Crossed Products and NC Dimensions Toronto, August 4 7 / 12

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 = AbelianoZ(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) Crossed Products and NC Dimensions Toronto, August 4 8 / 12

Non-free �ows

Theorem (Hirshberg-W)

R y X loc. cpt Hausd. with dim(X) <∞ ⇒ dimnuc(C0(X)oR) <∞.

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)oR→ C0(X)oR→ C0(X≤R)oR→ 0

Fact: dimnuc <∞ passes through extensions ⇒ Look at the two ends!1 R y X≤R is well-behaved (in particular, X≤R/R is Hausdor�) ⇒

dim+1nuc(C0(X≤R)oR) ≤ 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), R y X>R behaves like a free action for the purposeof the approximation ⇒ We mimic the approach for free actions.

Jianchao Wu (Penn State) Crossed Products and NC Dimensions Toronto, August 4 9 / 12

Application: C∗-algebras for 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)

Fact: each leaf of a line foliation ∼= R or S1.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) Crossed Products and NC Dimensions Toronto, August 4 10 / 12

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) Crossed Products and NC Dimensions Toronto, August 4 11 / 12

Thank you!

Jianchao Wu (Penn State) Crossed Products and NC Dimensions Toronto, August 4 12 / 12