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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 Nuclear dimension 1 Rokhlin Dimension 2 Topological actions: dim DAD , dim amen , and dim f . amen 3 Beyond free actions 4 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. dim nuc ( 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: id � A A . ψ φ := � d l =0 φ ( l ) B If no such d exists, we set dim nuc ( A ) = ∞ . dim nuc ( A ) = 0 ⇐ ⇒ A is AF ( = lim → ( fin.dim. C ∗ -alg ) ). − X topological space ⇒ dim nuc ( C 0 ( X )) = dim( X ) (covering dim.). A Kirchberg algebra (e.g. O n ) = ⇒ dim nuc ( A ) ≤ 1 . X metric space ⇒ dim nuc ( C ∗ u ( X )) ≤ as–dim( X ) (asymptotic dim.). Jianchao Wu (M¨ unster) Crossed product C ∗ -algebras and nuclear dimension Aug 17, 2015 3 / 20 dim nuc ( A ) < ∞ = ⇒ A is nuclear.

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: dim nuc (lim → A α ) ≤ lim inf dim nuc ( A α ) . − 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 dim nuc ( A ) < ∞ & G � A , when dim nuc ( 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 Nuclear dimension 1 Rokhlin Dimension 2 Topological actions: dim DAD , dim amen , and dim f . amen 3 Beyond free actions 4 Jianchao Wu (M¨ unster) Crossed product C ∗ -algebras and nuclear dimension Aug 17, 2015 5 / 20

Rokhlin Dimension for Finite Groups and Z m introduced by Hirshberg-Winter-Zacharias for finite groups and Z ; generalized to Z m 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 nuc ( A ) · dim fin , +1 dim +1 nuc ( A ⋊ α G ) ≤ dim +1 Rok ( α ) . Theorem (Toms-Winter, Hirshberg-Winter-Zacharias) nuc ( A ) · dim cyc , +1 dim +1 nuc ( A ⋊ α Z ) ≤ 2 · dim +1 ( α ) . Rok Theorem (Szabo) nuc ( A ⋊ α Z m ) ≤ 2 m · dim +1 nuc ( A ) · dim cyc , +1 dim +1 ( α ) . Rok 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 } ) { ( a n ) n ∈ N ∈ l ∞ ( N , A ) | [ a n , a ] → 0 , ∀ a ∈ A} F ∞ ( A ) := { ( a n ) n ∈ N ∈ l ∞ ( N , A ) | a n · a → 0 , a · a n → 0 , ∀ a ∈ A} is Kirchberg’s central sequence algebra . � α ∞ : G � F ∞ ( A ) . Definition (Szabo-W-Zacharias, after Hirshberg-Winter-Zacharias) The Rokhlin dimension dim Rok ( 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´ o-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( � Z m ) = 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 dim Rok ( α ) (Szabo-W-Zacharias) A ∼ = A ⊗ Z = ⇒ dim Rok ( α ) ≤ 1 is generic. A ∼ = A ⊗ Q = ⇒ dim Rok ( α ) ≤ 0 is generic. Here genericity means forming a dense G δ -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 Nuclear dimension 1 Rokhlin Dimension 2 Topological actions: dim DAD , dim amen , and dim f . amen 3 Beyond free actions 4 Jianchao Wu (M¨ unster) Crossed product C ∗ -algebras and nuclear dimension Aug 17, 2015 9 / 20

Amenability Dimension α X : (locally) compact Hausdorff, G � X continuously. Definition: Amenability Dimension α : G � X is said to have amenability dimension d (write: dim amen ( 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 α g ( x ) ∈ U ( l ) , then φ ( l ) ( α g ( x )) = g · φ ( l ) ( x ) . Examples dim amen ( G � βG ) = as–dim( G ) . irrational rotation: dim amen ( Z � S 1 ) = 2 . α dim amen ( α ) < ∞ = ⇒ G � X is a free and amenable action. ∃ relative version: dim amen ( α |F ) , where F : a collection of subgroups � � of 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 , dim amen ( 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 dim amen ( 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: dim DAD ( 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 = g n . . . g 1 | g n , . . . , g 1 ∈ M and g k . . . g 1 x ∈ U ( l ) , ∀ k ∈ { 1 , . . . , n } is finite. dim DAD ( α ) ≤ dim amen ( α ) . dim DAD can be defined for groupoids. Theorem (Guentner-Willett-Yu) ⇒ dim +1 nuc ( C ( X ) ⋊ α G ) ≤ dim +1 DAD ( α ) · dim +1 ( X ) . G � X compact = 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: dim f . 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 = ⇒ dim nuc ( C 0 ( X ) ⋊ α G ) ≤ dim f . 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 dim amen 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