Rocky Mountain Mathematics Consortium Summer School University of Wyoming, Laramie Large Subalgebras and the Structure of Crossed Products, Lecture 1: Introduction, Motivation, and the 1–5 June 2015 Cuntz Semigroup 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 1 June 2015 Comparison. 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: Introduction 1 June 2015 1 / 34 N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 2 / 34 A rough outline of all five lectures Introduction Introduction: what large subalgebras are good for. Main references: Definition of a large subalgebra. N. C. Phillips, Large subalgebras , preprint (arXiv: 1408.5546v1 Statements of some theorems on large subalgebras. [math.OA]). A very brief survey of the Cuntz semigroup. D. Archey and N. C. Phillips, Permanence of stable rank one for Open problems. centrally large subalgebras and crossed products by minimal Basic properties of large subalgebras. homeomorphisms , preprint (arXiv: 1505.00725v1 [math.OA]). A very brief survey of radius of comparison. T. Hines, N. C. Phillips, and A. S. Toms, Mean dimension and radius Description of the proof that if B is a large subalgebra of A , then A of comparison for minimal homeomorphisms with Cantor factors , in and B have the same radius of comparison. preparation. A very brief survey of crossed products by Z . Orbit breaking subalgebras of crossed products by minimal N. C. Phillips, Large subalgebras and applications , lecture notes. homeomorphisms. The first four lectures are mostly from the first paper, with a small amount Sketch of the proof that suitable orbit breaking subalgebras are large. of material from the second paper. The last lecture is from the third paper. A very brief survey of mean dimension. The proof of the result in the third lecture is quite different from that in Description of the proof that for minimal homeomorphisms with the first paper. The lecture notes contain a substantial amount of material Cantor factors, the radius of comparison is at most half the mean not in the actual lectures, but condensed considerably from the first paper. dimension. N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 3 / 34 N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 4 / 34
Applications Applications (continued) The first large subalgebra was used by Putnam in 1989 (not by name) to From the previous slide: Large subalgebras are used to prove that if study the order on K 0 ( C ∗ ( Z , X , h )) when h is a minimal homeomorphism h : X → X is a minimal homeomorphism and X has a surjective map to the Cantor set, then rc( C ∗ ( Z , X , h )) ≤ 1 of the Cantor set. They have been used in a number of places (still 2 mdim( h ). without the name) to study the structure of crossed products by minimal We also show that for minimal homeomorphisms of the type considered by homeomorphisms. (Some references are in the notes.) The main recent Giol and Kerr, we actually have rc( C ∗ ( Z , X , h )) = 1 2 mdim( h ). uses are as follows: 1 The “extended” irrational rotation algebras, obtained by “cutting” The applications to C ∗ ( Z , X , h ) use the “orbit breaking subalgebra” each of the standard unitary generators at one or more points in its C ∗ ( Z , X , h ) Y (defined below). Other applications (such as the first proof spectrum, are AF (Elliott-Niu). that if Z d acts freely and minimally on a finite dimensional compact 2 If h : X → X is a minimal homeomorphism of an infinite compact metric space, then C ∗ ( Z d , X ) has strict comparison of positive elements) metric space with mean dimension zero, then C ∗ ( Z , X , h ) is Z -stable require large subalgebras for which we don’t have a formula, only an (Elliott-Niu). existence proof. (We won’t get to such examples in this course.) 3 If h : X → X is a minimal homeomorphism and X has a surjective map to the Cantor set K , then C ∗ ( Z , X , h ) has stable rank one, The result on C ∗ ( Z d , X ) has been superseded by Rokhlin dimension regardless of the mean dimension of h (joint with Archey). methods. There unfortunately is no time in this course to say anything 4 If h : X → X is a minimal homeomorphism and X has a surjective about Rokhlin dimension, but in many problems one should consider both map to K , then rc( C ∗ ( Z , X , h )) ≤ 1 2 mdim( h ) (with Hines and Toms). Rokhlin dimension and large subalgebras as possible methods. N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 5 / 34 N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 6 / 34 About the definitions Definition a � A b if there is a sequence ( v n ) ∞ n =1 in K ⊗ A such that v n bv ∗ Let A be a C*-algebra, and let a , b ∈ ( K ⊗ A ) + . We say that a is Cuntz n → a . subequivalent to b over A , written a � A b , if there is a sequence ( v n ) ∞ n =1 More about Cuntz comparison later. in K ⊗ A such that lim n →∞ v n bv ∗ n = a . From the previous slide: A unital subalgebra B ⊂ A is large in A if for Definition a 1 , a 2 , . . . , a m ∈ A , ε > 0, x ∈ A + with � x � = 1, and y ∈ B + \ { 0 } , there Let A be an infinite dimensional simple unital C*-algebra. A unital are c 1 , c 2 , . . . , c m ∈ A and g ∈ B such that: subalgebra B ⊂ A is said to be large in A if for every m ∈ Z > 0 , 1 0 ≤ g ≤ 1. a 1 , a 2 , . . . , a m ∈ A , ε > 0, x ∈ A + with � x � = 1, and y ∈ B + \ { 0 } , there 2 For j = 1 , 2 , . . . , m we have � c j − a j � < ε . are c 1 , c 2 , . . . , c m ∈ A and g ∈ B such that: 3 For j = 1 , 2 , . . . , m we have (1 − g ) c j ∈ B . 1 0 ≤ g ≤ 1. 2 For j = 1 , 2 , . . . , m we have � c j − a j � < ε . 4 g � B y and g � A x . 5 � (1 − g ) x (1 − g ) � > 1 − ε . 3 For j = 1 , 2 , . . . , m we have (1 − g ) c j ∈ B . 4 g � B y and g � A x . B being unital means 1 A ∈ B . 5 � (1 − g ) x (1 − g ) � > 1 − ε . The Cuntz subequivalence involving y in (4) is relative to B , not A . N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 7 / 34 N. C. Phillips (U of Oregon) Large Subalgebras: Introduction 1 June 2015 8 / 34
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