Rokhlin Dimension Beyond Residually Finite Groups

Transcription

1 Rokhlin Dimension Beyond Residually Finite Groups Jianchao Wu Penn State University UAB, July 12, 2017 Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 1 / 9

2 Nuclear dimension and the Elliott classication program Winter and Zacharias developed a kind of dimension theory for (nuclear) C -algebras. dim nuc : CStarAlg Z 0 { }. Some basic properties: X topological space dim nuc (C 0 (X)) = dim(x) (covering dim.). X a metric space dim nuc (Cu(X)) asdim(x) (asymptotic dim.). dim nuc (A) = 0 A is AF (= lim(n.dim. C -alg)). A Kirchberg algebra (e.g. O n ) = dim nuc (A) = 1. Finite nuclear dimension is preserved under taking:,, quotients, hereditary subalgebras, direct limits, extensions, etc. Theorem (Elliott-Gong-Lin-Niu, G-L-N, Tikuisis-White-Winter,..., Kirchberg-Phillips,... ) The class of unital simple separable C -algebras with nite nuclear dimension (FAD) and satisfying UCT is classied by the Elliott invariant. Since many fun examples are constructed as crossed products, we ask: Question: When does FAD pass through taking crossed products? More precisely, if dim nuc (A) < & G A, when dim nuc (A G) <? Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 2 / 9

3 dim nuc (A) < ¾when? = dim nuc (A G) < In general, this is dicult. But there are some successful strategies, e.g., using dimensions dened for actions. First case: If A = C(X), there are a number of closely-related dimensions that we can use, e.g., dynamical asymptotic dimension (Guentner-Willett-Yu), amenability dimension (G-W-Y), (ne) tower dimension (Kerr). We will focus on tower dimension in this talk. Let X be a compact Hausdor space and α: G X an action. Denition (Tower dimension) A tower is a pair (B, S) where B X is open, S G, and {α g (B): g S} is disjoint. dim tow (α) d i F G, towers (B i, S i ) for i I = I (0)... I (d), such that 1 l {0,..., d}, {α g (B i ): g S i, i I (l) } is disjoint; 2 x X, i I and g S i such that F g S i and x α g (B i ) (= X = g S i,i I α g(b i )). Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 3 / 9

4 Denition (Tower dimension) dim tow (α) d i F G, towers (B i, S i ) for i I = I (0)... I (d), such that 1 l {0,..., d}, {α g (B i ): g S i, i I (l) } is disjoint; 2 x X, i I and g S i such that F g S i and x α g (B i ). Theorem (Kerr) nuc(c(x) α G) (X) tow (α). If G is nite, we may choose F = G. Then WLOG, S i = G, i, and each tower is a local trivialization of the action. The tower dimension recovers a notion called the G-index. Example: The odometer action α: Z Z n has dim tow (α) = 1. More generally, if G is residually nite and lim G/G i is its pronite completion assoc. to a separating chain G 1 > G 2 >... of nite-index subgroups, we have dim tow (G lim G/G i ) asdim( {Gi }G), the asymptotic dimension of the box space assoc. to {G i }. dim tow (G βg) = asdim(g). E.g., this works for the free group. Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 4 / 9

5 Theorem (Kerr) nuc(c(x) α G) (X) tow (α). Ingredients in the proof: A tower (B, S) determines a subalgebra of C(X) G: {u g fu h : f C 0(B), h, g S} = M S (C 0 (X)). The F -overlaps of the towers help produce a partition of unity subject to the towers consisting of functions which are almost at along orbits. Thus we would like to verify dim am ( ) <. Theorem (Szabo-W-Zacharias) Let G be a f.g. nilpotent group and G α X a free action on a compact metric space with dim(x) <. Then dim tow (α) <. Corollary For such a (X, G, α), the crossed product C(X) α G has nite nuclear dimension and is thus in the class of simple separable unital nuclear C -algebras that are classiable by the Elliott invariant. Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 5 / 9

6 For C -dynamical systems G A, one strategy is the Rokhlin dimension. This was dened rst for Z and for nite groups by Hirshberg, Winter and Zacharias. G: countable discrete residually nite group, and α : G A. {G i } i=1 : a sequence of nite-index subgroups s.t. i=1 G i = {1}. F (A) is Kirchberg's central sequence algebra. When A is unital, F (A) := (l (N, A)/C 0 (N, A)) {constant sequences}. α α : G F (A). A c.p. map φ : A B is called order-zero i aa = 0 = φ(a)φ(a ) = 0, a, a A (e.g. -homomorphisms are order-zero). Denition (Szabo-W-Zacharias, after Hirshberg-Winter-Zacharias) The Rokhlin dimension of α w.r.t {G i }, written as dim Rok (α, {G i }) = d, is the smallest d N s.t. for every G i, equivariant c.p.c. order-zero maps φ (l) : (C(G/G i ), G-shift) (F (A), α ) for l = 0,, d, s.t. φ (0) (1) + + φ (l) (1) = 1. Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 6 / 9

7 G r.f. box space G. Writing ( ) := dim( ) + 1, we have: Theorem (Szabó-W-Zacharias) nuc(a α,w G) as ( G) nuc(a) Rok (α). Theorem (Delabie-Tointon, after Szabó-W-Zacharias, Finn-SellW) If G is virtually nilpotent, then asdim( G) = asdim(g) (= Hirsch length(g)). For residually nite G, there are close relations between the two notions: Theorem (Szabó-W-Zacharias) Rok (α) dim tow(α) as ( G) Rok (α). Drawback: The requirement that asdim( G) < limits us to certain residually nite amenable groups. Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 7 / 9

8 The theorem of Kerr generalizes to G-X-C -algebras. E.g., when B = A C(X) with G acting diagonally by α β, we have nuc((a C(X)) α β G) nuc(a) (X) tow (β). For general noncommutative A, we may dene a relative notion: Denition (Hamblin-W-Zacharias, after Hirshberg-Winter-Zacharias): dim Rok (α β), the Rokhlin dimension of α: G A w.r.t β : G X dim Rok (α β) d i equivariant c.p. contractive order-zero maps φ (0),..., φ (d) : (C(X), β) (F (A), α), s.t. d l=0 φ(l) (1) = 1. F (A) is Kirchberg's central sequence algebra. When A is unital, F (A) := (l (N, A)/C 0 (N, A)) {constant sequences}. Order-zero = orthogonality preserving. Remark: C(X) may be replaced by other unital C -algebras. Theorem (Hamblin-W-Zacharias) nuc(a α,w G) nuc(a) tow (β) dim+1 Rok (α β). This generalizes the theory for residually nite groups G (e.g., Z): given a separating chain of nite-index subgroups {G i }, we take X = limg/g i, the pronite completion = dim am (β) asdim( {Gi }G). Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 8 / 9

9 For r.f. groups, take X = lim G/G i = dim am (β) asdim( {Gi }G). We recover: Theorem (Szabó-W-Zacharias) nuc(a α,w G) nuc(a) Rok (α) asdim+1 ( G). Jianchao Wu (Penn State) Rokhlin dimension beyond r.f. groups UAB, July 12 9 / 9