Crossed product C -algebras and nuclear dimension Jianchao Wu - - PowerPoint PPT Presentation

Crossed product C∗-algebras and nuclear dimension

Jianchao Wu

University of M¨ unster

Aug 17, 2015

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 1 / 20

Table of Contents

1

Nuclear dimension

2

Rokhlin Dimension

3

Topological actions: dimDAD, dimamen, and dimf.amen

4

Beyond free actions

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 2 / 20

Nuclear Dimension: dimension theory for (nuclear) C∗-alg.

Definition: Nuclear Dimension (Winter-Zacharias)

Let A be a C∗-algebra. dimnuc(A) is defined to be the smallest d ∈ N s.t. ∀F ⊂fin A, ∀ǫ > 0, ∃ a fin-dim’l alg. B, a completely positive contractive (c.p.c.) ψ : A → B and c.p.c. order-zero maps φ(0), . . . , φ(d) : B → A, s.t. the following diagram commutes on F with errors ≤ ǫ in norm: A

ψ

• id

A

B

φ := d

l=0 φ(l)

• .

If no such d exists, we set dimnuc(A) = ∞. dimnuc(A) = 0 ⇐ ⇒ A is AF (= lim − →(fin.dim. C∗-alg)). X topological space ⇒ dimnuc(C0(X)) = dim(X) (covering dim.). A Kirchberg algebra (e.g. On) = ⇒ dimnuc(A) ≤ 1. X metric space ⇒ dimnuc(C∗

u(X)) ≤ as–dim(X) (asymptotic dim.).

dimnuc(A) < ∞ = ⇒ A is nuclear.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 3 / 20

Finite Nuclear Dimensions and Permanence Properties

Conjecture (Toms-Winter): equivalence of regularity properties

For a nuclear unital simple separable C∗-algebra A, TFAE: A has finite nuclear dimension (FND). A is Z-stable (A ∼ = A ⊗ Z, Z being the Jiang-Su algebra). A has strict comparison. FND is preserved under taking: ⊕, ⊗, quotients, hereditary subalgebras, direct limits, extensions, etc — with formulas like: Direct limit: dimnuc(lim

− → Aα) ≤ lim inf dimnuc(Aα).

Tensor product: dim+1

nuc(A ⊗ B) ≤ dim+1 nuc(A) · dim+1 nuc(B).

A more interesting question

When does finite nuclear dimension pass through taking crossed products? More precisely, if dimnuc(A) < ∞ & G A, when dimnuc(A ⋊ G) < ∞? Strategy: study the complexity of the action, e.g. in terms of a dimension.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 4 / 20

Table of Contents

1

Nuclear dimension

2

Rokhlin Dimension

3

Topological actions: dimDAD, dimamen, and dimf.amen

4

Beyond free actions

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 5 / 20

Rokhlin Dimension for Finite Groups and Zm

introduced by Hirshberg-Winter-Zacharias for finite groups and Z; generalized to Zm by Szabo; also works for compact groups (Hirshberg-Phillips, Gardella...); measures the complexity of the action in terms of decomposition.

Theorem (Hirshberg-Winter-Zacharias)

If G is a finite group, then dim+1

nuc(A ⋊α G) ≤ dim+1 nuc(A) · dimfin,+1 Rok (α).

Theorem (Toms-Winter, Hirshberg-Winter-Zacharias)

dim+1

nuc(A ⋊α Z) ≤ 2 · dim+1 nuc(A) · dimcyc,+1 Rok

(α).

Theorem (Szabo)

dim+1

nuc(A ⋊α Zm) ≤ 2m · dim+1 nuc(A) · dimcyc,+1 Rok

(α).

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 6 / 20

Rokhlin Dimension for Residually Finite Groups

G: countable discrete r.f. group, and α : G A. (Recall: r.f. = residually finite: {finite index subgroups of G} = {1}) F∞(A) := {(an)n∈N ∈ l∞(N, A) | [an, a] → 0, ∀a ∈ A} {(an)n∈N ∈ l∞(N, A) | an · a → 0, a · an → 0, ∀a ∈ A} is Kirchberg’s central sequence algebra. α∞ : G F∞(A).

Definition (Szabo-W-Zacharias, after Hirshberg-Winter-Zacharias)

The Rokhlin dimension dimRok(α) is the smallest d ∈ N s.t. for every finite index normal subgroup H < G, ∃ equivariant c.p.c. order zero maps φ(l) : (C(G/H), G-shift) → (F∞(A), α∞) for l = 0, · · · , d, s.t. φ(0)(1) + · · · + φ(l)(1) = 1. This generalizes previous definitions. The rough idea is to approximately decompose the action α into (d + 1)-many shift actions on finite quotients.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 7 / 20

Theorem (Szab´

• -W-Zacharias)

dim+1

nuc(A ⋊α,w G) ≤ as–dim+1(G) · dim+1 nuc(A) · dim+1 Rok(α)

. as–dim(G) is the asymptotic dimension of the box space G of G. It captures the necessary coarse geometric information of the group. E.g. as–dim(Zm) = m. It is finite for a reasonably large class of groups:

Theorem (Finn-Sell–W)

If G is elementary amenable, then as–dim(G) ≤ Hirsch length of G. In particular, as–dim(G) < ∞ if G is (locally finite by) virtually polycyclic.

Genericity Theorems for dimRok(α) (Szabo-W-Zacharias)

A ∼ = A ⊗ Z = ⇒ dimRok(α) ≤ 1 is generic. A ∼ = A ⊗ Q = ⇒ dimRok(α) ≤ 0 is generic. Here genericity means forming a dense Gδ-subset in the “space of actions” with topology of point-wise limits, and Q is the univeral UHF algebra. Later we will discuss the case when A is commutative.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 8 / 20

Table of Contents

1

Nuclear dimension

2

Rokhlin Dimension

3

Topological actions: dimDAD, dimamen, and dimf.amen

4

Beyond free actions

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 9 / 20

Amenability Dimension

X: (locally) compact Hausdorff, G

α

X continuously.

Definition: Amenability Dimension

α : G X is said to have amenability dimension d (write: dimamen(α) = d) if d is the smallest natural number s.t. ∀ M ⊂fin G, ∃ an open cover U = {U (0), . . . , U (d)} of X and continuous maps φ(l) : U (l) → G, l = 0, . . . , d, that are M-equivariant: ∀ x ∈ U (l), g ∈ M, if αg(x) ∈ U (l), then φ(l)(αg(x)) = g · φ(l)(x).

Examples

dimamen(G βG) = as–dim(G). irrational rotation: dimamen(Z S1) = 2. dimamen(α) < ∞ = ⇒ G

α

X is a free and amenable action. ∃ relative version: dimamen(α|F), where F: a collection of subgroups

• f G closed under conjugation, e.g. F =
• {1}
• , Fin, Cyc, VCyc, etc.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 10 / 20

Discourse: applications to Farrell-Jones and Baum-Connes

Farrell-Jones conjecture: how to compute K∗(R[G]) if we know “K∗(R[ virtually cyclic subgroups of G ])”? (R: G-ring)

Theorem (Bartels-L¨ uck-Reich)

The Farrell-Jones conjecture holds for hyperbolic groups (with any R). An important and painstaking step in the proof is (equivalent to) showing that ∀ hyperbolic G, dimamen(G ∂G | VCyc) < ∞, where ∂G is the Gromov boundary of the Cayley graph of G. Baum-Connes conjecture: how to compute K∗(A ⋊ G) if we know “K∗(A ⋊ ( finite subgroups of G ))”? (A: G-C*-algebra)

Theorem (Guentner-Willett-Yu)

If dimamen(G X | Fin) < ∞, then the Baum-Connes conjecture holds with coefficient C(X) ⊗ A, for any G-C∗-alg A. The proof does not use “transendental methods”. By a standard argument, this implies the Novikov conjecture for G.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 11 / 20

Dynamic Asymptotic Dimension

Definition (Guentner-Willett-Yu): Dynamic Asymptotic Dimension

α : G X is said to have dynamic asymptotic dimension d (write: dimDAD(α) = d) if d is the smallest natural number s.t. ∀ M ⊂fin G, ∃ an open cover U = {U (0), . . . , U (d)} of X s.t. ∀ l ∈ {0, . . . , d}, ∀ x ∈ U (l), the set

• g = gn . . . g1 | gn, . . . , g1 ∈ M and gk . . . g1x ∈ U (l), ∀k ∈ {1, . . . , n}
• is finite.

dimDAD(α) ≤ dimamen(α). dimDAD can be defined for groupoids.

Theorem (Guentner-Willett-Yu)

G X compact = ⇒ dim+1

nuc(C(X) ⋊α G) ≤ dim+1 DAD(α) · dim+1(X).

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 12 / 20

Fine Amenability Dimension

Definition: Fine Amenability Dimension

α : G X is said to have fine amenability dimension d (write: dimf.amen(α) = d) if d is the smallest natural number s.t. ∀ M ⊂fin G, ∀ finite open cover V of X, ∃ an open cover U = {U (0), . . . , U (d)} of X that refines V, and continuous maps φ(l) : U (l) → G, l = 0, . . . , d, that are M-equivariant.

Theorem

G X (locally) compact = ⇒ dimnuc(C0(X) ⋊α G) ≤ dimf.amen(α).

Intertwining inequality = ⇒ equiv. of finiteness when dim(X) < ∞

dim+1

DAD(α) ≤ dim+1 amen(α) ≤ dim+1 f.amen(α) ≤ dim+1 DAD(α) · dim+1(X).

Remark: another way to strengthen dimamen due to Kerr

The (fine) tower dimension dim(f)tow(α) = ⇒ fit into a similar intertwining inequality.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 13 / 20

Relation to Rokhlin dimension

For residually finite G, there are close relations between dimamen and dimRok:

Theorem (Szab´

• -W-Zacharias)

dim+1

Rok(α) ≤ dimamen(α) ≤ as–dim+1(G) · dim+1 Rok(α)

. Thus, when as–dim(G) < ∞ (e.g. G is locally finite by virtually polycyclic), we have dimRok(α) < ∞ ⇐ ⇒ dimRok(α) < ∞ and the two approaches to showing dimnuc(C0(X) ⋊ G) < ∞ agree. = ⇒ Suggests: a unifying approach: amenability dimension for C∗-dynamics (in prog.).

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 14 / 20

The case of R

For R

α

X, we can define tube dimension dimtube(α), which is similar to dimtow and dimamen in the discrete case. It has close relations with the Rokhlin dimension for the induced C∗-flow R

α

C(X).

Theorem (Hirshberg-Szab´

• -W-Winter)

dim+1

Rok(α) ≤ dim+1 tube(α) ≤ 2 dim+1 Rok(α)

. One can also estimate dim+1

nuc(C(X) ⋊α R) directly with the help of

dimtube(α).

Theorem (Hirshberg-Szab´

• -W-Winter)

dim+1

nuc(C(X) ⋊α R) ≤ dim(X) · dim+1 tube(α)

.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 15 / 20

How common is dimamen(α) < ∞?

We mentioned: dimamen(G ∂G | VCyc) < ∞ for G hyperbolic (BLR).

Theorem (Szabo-W-Zacharias)

Let G be a f.g. nilpotent group and G

α

X a free continuous action on a compact metric space with dim(X) < ∞. Then dimamen(α) < ∞. The proof uses the marker property (Gutman, Szabo).

Corollary

In this case, also dimRok(α) < ∞ and dimnuc(C(X) ⋊α G) < ∞.

Theorem (Bartels-L¨ uck-Reich, essentially)

Let R

α

X be a free flow on a compact metric space with dim(X) < ∞. Then dimtube(α) < ∞.

Theorem (Hirshberg-Szab´

• -W-Winter)

In this case, also dimRok(α) < ∞ and dimnuc(C(X) ⋊α R) < ∞.

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 16 / 20

Table of Contents

1

Nuclear dimension

2

Rokhlin Dimension

3

Topological actions: dimDAD, dimamen, and dimf.amen

4

Beyond free actions

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 17 / 20

Non-free actions for one-parameter group actions

Theorem (Hirshberg-W)

Let Z (resp. R)

α

X be a continuous action on a locally compact Hausdorff space with dim(X) < ∞. Then dimnuc(C(X) ⋊α Z) < ∞

• resp. dimnuc(C(X) ⋊α R) < ∞
• .

New examples of C∗-algebras with finite nuclear dimensions

dimnuc(C∗(Z2 ⋊A Z)) < ∞, where A = 2 1 1 1

• ∈ SL(2, Z).

This group is polycyclic but not nilpotent. Thm(Eckhardt-McKenney): dimnuc(C∗(any f.g. nilpotent grp)) < ∞.

dimnuc(C∗(L)) < ∞ for L = Z2 ≀ Z = Z

Z 2

⋊shift Z (lamplighter grp).

dr(C∗(L)) = ∞ because of lack of strong QD = ⇒ a C∗-algebra with finite dimnuc, infinite decomposition rank and a faithful trace. C∗(L) has exponential growth (w.r.t. the group generators).

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 18 / 20

Questions

The proof of the last theorem is a bit ad-hoc, so can we develop a general strategy (similar to dimamen) to treat other G? In particular,

Question

Let G

α

X be a continuous action on a compact metric space with dim(X) < ∞. Let F be a collection of nilpotent subgroups of G that

is closed under conjugation and taking subgroups, and satisfies sup{dimnuc(C∗(H)) |H ∈ F} < ∞.

Assume dimamen(α|F) < ∞. Does it follow that dimnuc(C(X) ⋊α G) < ∞? So far we have more tools to find upper bounds than lower bounds.

Question

Can we develop machineries to estimate the lower bounds for dimnuc, dimamen, etc, e.g. through homological methods?

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 19 / 20

Summary

There is a family of new invariants / regularity properties for topological dynamics. They come in the form of dimensions, which measure the complexity

• f decomposing the dynamical system into simpler pieces.

Since such dimensionally-controlled decompositions facilitate computations of K-theory and estimations of nuclear dimensions, these new dimensions are useful in

finding (conceptually unified) approaches to Farrell-Jones and Baum-Connes conjectures, and verifying regularity properties for crossed product C∗-algebras (which may in return help the study of topological dynamics).

These dimensions are finite for some reasonably large classes of systems. There is still a lot to be done!

Thank you!

Jianchao Wu (M¨ unster) Crossed product C∗-algebras and nuclear dimension Aug 17, 2015 20 / 20