Large Subalgebras and the Structure of Crossed Products, Lecture 5: - PowerPoint PPT Presentation

Rocky Mountain Mathematics Consortium Summer School University of Wyoming, Laramie Large Subalgebras and the Structure of Crossed Products, Lecture 5: Application to the Radius of 1–5 June 2015 Comparison of Crossed Products by Minimal Homeomorphisms Lecture 1 (1 June 2015): Introduction, Motivation, and the Cuntz Semigroup. N. Christopher Phillips Lecture 2 (2 June 2015): Large Subalgebras and their Basic Properties. University of Oregon Lecture 3 (4 June 2015): Large Subalgebras and the Radius of Comparison. 5 June 2015 Lecture 4 (5 June 2015 [morning]): Large Subalgebras in Crossed Products by Z . Lecture 5 (5 June 2015 [afternoon]): Application to the Radius of Comparison of Crossed Products by Minimal Homeomorphisms. N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 1 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 2 / 24 A rough outline of all five lectures Reminder: Covering dimension Introduction: what large subalgebras are good for. Definition Definition of a large subalgebra. Statements of some theorems on large subalgebras. Let X be a compact Hausdorff space. A very brief survey of the Cuntz semigroup. 1 Let U be a finite open cover of X . The order of U is the least number Open problems. n such that the intersection of any n + 2 distinct elements of U is Basic properties of large subalgebras. � empty. (The formula ord( U ) = − 1 + sup x ∈ X U ∈U χ U ( x ) is often A very brief survey of radius of comparison. used.) Description of the proof that if B is a large subalgebra of A , then A 2 Let U and V be finite open covers of X . Then V refines U (written and B have the same radius of comparison. V ≺ U ) if for every V ∈ V there is U ∈ U such that V ⊂ U . A very brief survey of crossed products by Z . Orbit breaking subalgebras of crossed products by minimal 3 Let U be a finite open cover of X . We define the dimension D ( U ) to homeomorphisms. be the least possible order of a finite open cover which refines U . Sketch of the proof that suitable orbit breaking subalgebras are large. That is, D ( U ) = inf V≺U ord( V ). A very brief survey of mean dimension. 4 The covering dimension dim( X ) is the supremum of D ( U ) over all Description of the proof that for minimal homeomorphisms with finite open covers U of X . Cantor factors, the radius of comparison is at most half the mean dimension. N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 3 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 4 / 24

Covering dimension (continued) Mean dimension Recall the definitions involving covers from the previous slide: Let X be a compact metric space and let h : X → X be a 1 ord( U ) is the least number n such that the intersection of any n + 1 homeomorphism. (For best behavior, h should not have “too many” distinct elements of U is empty. periodic points.) Lindenstrauss and Weiss defined the mean dimension 2 V ≺ U if for every V ∈ V there is U ∈ U such that V ⊂ U . mdim( h ). It is designed so that if K is a sufficiently nice compact metric 3 D ( U ) = inf V≺U ord( V ). space (in particular, dim( K n ) should equal n · dim( K ) for all n ), then the 4 dim( X ) = sup U D ( U ). shift on X = K Z should have mean dimension equal to dim( K ). We can observe that if X is totally disconnected, then dim( X ) = 0. One Given this heuristic, it should not be surprising that if dim( X ) < ∞ then sees dim([0 , 1]) = 1 by using open covers consisting of short intervals each mdim( h ) = 0. of which only intersects its immediate neighbors. One sees Definition dim([0 , 1] 2 ) = 2 by using open covers consisting of small neighborhoods of Let X be a compact metric space, and let U and V be two finite open the tiles in a fine hexagonal tiling. It is harder to see what is happening in covers of X . Then the join U ∨ V of U and V is higher dimensions. � � U ∨ V = U ∩ V : U ∈ U and V ∈ V . It is a fact that dim( X × Y ) ≤ dim( X ) + dim( Y ), with equality if X and Y are sufficiently nice (for example, finite complexes). However, equality need not hold in general. N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 5 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 6 / 24 Definition of mean dimension Mean dimension and radius of comparison � � Recall: U ∨ V = U ∩ V : U ∈ U and V ∈ V . Definition Theorem from Lecture 1: Let X be a compact metric space and let h : X → X be a homeomorphism. Then the mean dimension of h is Theorem (Joint work with Hines and Toms) U ∨ h − 1 ( U ) ∨ · · · ∨ h − n +1 ( U ) � � D Let X be a compact metric space. Assume that there is a continuous mdim( h ) = sup lim . surjective map from X to the Cantor set. Let h : X → X be a minimal n n →∞ U homeomorphism. Then rc( C ∗ ( Z , X , h )) ≤ 1 2 mdim( h ). The supremum is over all finite open covers of X (just like in the definition It is hoped that rc( C ∗ ( Z , X , h )) = 1 of dim( X )). 2 mdim( h ) for any minimal homeomorphism of an infinite compact metric space X . We can show this The definition uses the join of n covers. One needs to prove that the limit for some special systems covered by this theorem, slightly generalizing the exists; we omit this. construction of Giol and Kerr. If dim( X ) < ∞ , then the numerator is always at most dim( X ), so the limit is zero. More generally, if h is minimal and has at most countably many ergodic measures, then mdim( h ) = 0. N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 7 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 8 / 24

Factor systems Cantor set factors Proposition The hypothesis on existence of a surjective map to the Cantor set has Let X be a compact metric space, and let h : X → X be a minimal other equivalent formulations, one of which is the existence of an equivariant surjective map to the Cantor set. We need a definition. homeomorphism. Then the following are equivalent: 1 There exists a decreasing sequence Y 0 ⊃ Y 1 ⊃ Y 2 ⊃ · · · of nonempty Definition compact open subsets of X such that the subset Y = � ∞ n =0 Y n Let h : X → X and k : Y → Y be homeomorphisms. We say that the satisfies h r ( Y ) ∩ Y = ∅ for all r ∈ Z \ { 0 } . system ( Y , k ) is a factor of ( X , h ) if there is a surjective continuous map 2 There is a minimal homeomorphism of the Cantor set which is a g : X → Y (the factor map ) such that g ◦ h = k ◦ g . factor of ( X , h ). 3 There is a continuous surjective map from X to the Cantor set. The requirement in the definition is that the following diagram commute: 4 For every n ∈ Z > 0 there is a partition P of X into at least n h X − − − − → X nonempty compact open subsets.   g � g   We omit the proof. � k To keep things simple, in this lecture we will assume that h has a − − − − → Y . Y particular minimal homeomorphism of the Cantor set as a factor, namely an odometer system. N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 9 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 10 / 24 Odometer as a factor system Odometers Definition Assume h is minimal and h n ( Y ) ∩ Y = ∅ for n ∈ Z \ { 0 } . Write Y = � ∞ Let d = ( d n ) n ∈ Z > 0 be a sequence in Z > 0 with d n ≥ 2 for all n ∈ Z > 0 . The n =0 Y n with Y 0 ⊃ Y 1 ⊃ · · · and int( Y n ) � = ∅ for all n ∈ Z ≥ 0 . Then C ∗ ( Z , X , h ) Y = lim → n C ∗ ( Z , X , h ) Y n , and C ∗ ( Z , X , h ) Y n is a recursive d-odometer is the minimal system ( X d , h d ) defined as follows. Set − subhomogeneous C*-algebra whose base spaces are closed subsets of X . ∞ � X d = { 0 , 1 , 2 , . . . , d n − 1 } , The effect of requiring a Cantor system factor is that one can choose Y n =1 and ( Y n ) n ∈ Z ≥ 0 so that Y n is both closed and open for all n ∈ Z ≥ 0 . This which is homeomorphic to the Cantor set. For x = ( x n ) n ∈ Z > 0 ∈ X d , let ensures that C ∗ ( Z , X , h ) Y n is a homogeneous C*-algebra whose base �� �� n 0 = inf n ∈ Z > 0 : x n � = d n − 1 . spaces are closed subsets of X . Thus C ∗ ( Z , X , h ) Y is a simple AH algebra. If n 0 = ∞ set h d ( x ) = (0 , 0 , . . . ). Otherwise, h d ( x ) = ( h d ( x ) n ) n ∈ Z > 0 is This is done by taking Y to be the inverse image of a point in the Cantor  set. 0 n < n 0   h d ( x ) n = x n + 1 n = n 0 The further simplification of assuming an odometer factor (definition on  next slide) is that one can arrange C ∗ ( Z , X , h ) Y n ∼ x n n > n 0 .  = M p n ( C ( Y n )), that is, there is only one summand. This simplifies the notation but otherwise It is “addition of (1 , 0 , 0 , . . . ) with carry to the right”. When n 0 � = ∞ , we makes little difference. have � � h ( x ) = 0 , 0 , . . . , 0 , x n 0 + 1 , x n 0 +1 , x n 0 +2 , . . . . N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 11 / 24 N. C. Phillips (U of Oregon) Large Subalgebras: Minimal Homeomorphisms 5 June 2015 12 / 24