Large Subalgebras and the Structure of Crossed Products, Lecture 4: - PowerPoint PPT Presentation

Rocky Mountain Mathematics Consortium Summer School University of Wyoming, Laramie Large Subalgebras and the Structure of Crossed Products, Lecture 4: Large Subalgebras in Crossed 1–5 June 2015 Products by Z 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 5 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: Crossed Products by Z 5 June 2015 1 / 28 N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 2 / 28 A rough outline of all five lectures A brief reminder on crossed products Let G be a (discrete) group, let A be a unital C*-algebra, and let Introduction: what large subalgebras are good for. α : G → Aut( A ) be an action of G on A . Definition of a large subalgebra. Statements of some theorems on large subalgebras. The skew group ring A [ G ] is the set of all formal sums A very brief survey of the Cuntz semigroup. � Open problems. a g u g Basic properties of large subalgebras. g ∈ G A very brief survey of radius of comparison. with a g ∈ A for all g ∈ G and a g = 0 for all but finitely many g ∈ G . The Description of the proof that if B is a large subalgebra of A , then A product and adjoint are determined by requiring that: and B have the same radius of comparison. u g is unitary for g ∈ G . A very brief survey of crossed products by Z . u g u h = u gh for g , h ∈ G . Orbit breaking subalgebras of crossed products by minimal u g au ∗ g = α g ( a ) for g ∈ G and a ∈ A . homeomorphisms. Thus, Sketch of the proof that suitable orbit breaking subalgebras are large. ( a · u g )( b · u h ) = ( a [ u g bu g − 1 ]) · u gh = ( a α g ( b )) · u gh A very brief survey of mean dimension. ( a · u g ) ∗ = α − 1 g ( a ∗ ) · u g − 1 Description of the proof that for minimal homeomorphisms with Cantor factors, the radius of comparison is at most half the mean for a , b ∈ A and g , h ∈ G , extended linearly. dimension. N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 3 / 28 N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 4 / 28

As above, G is a discrete group, A is a unital C*-algebra, and Conditional expectation and coefficients α : G → Aut( A ) is an action of G on A . As above, G is a discrete group, A is a unital C*-algebra, and Fix a faithful representation π : A → L ( H 0 ) of A on a Hilbert space H 0 . α : G → Aut( A ) is an action of G on A . Set H = l 2 ( G , H 0 ), the set of all ξ = ( ξ g ) g ∈ G in � g ∈ G H 0 such that As above, C ∗ r ( G , A , α ) is the completion of A [ G ] in the norm coming from g ∈ G � ξ g � 2 < ∞ , with the scalar product � a representation modelled on the regular representation of G . � � � ( ξ g ) g ∈ G , ( η g ) g ∈ G = � ξ g , η g � . The full crossed product C ∗ ( G , A , α ) is in principle bigger, but for g ∈ G amenable groups, including Z , it is known to be the same. In this case, we Then define σ : A [ G ] → L ( H ) as follows. For a = � g ∈ G a g u g , just write C ∗ ( G , A , α ). � π ( α − 1 If 1 is the identity of the group, then a �→ au 1 is an injective unital ( σ ( a ) ξ ) h = h ( a g ))( ξ g − 1 h ) homomorphism from A to C ∗ r ( G , A , α ). Identify A with its image. g ∈ G for all h ∈ G . In particular, ( σ ( u g ) ξ ) h = ξ g − 1 h for g ∈ G and The map � g ∈ G a g u g �→ a 1 , from A [ G ] to A ⊂ A [ G ], extends to a faithful conditional expectation E : C ∗ r ( G , A , α ) → A ⊂ C ∗ ( σ ( au 1 ) ξ ) h = π ( α − 1 r ( G , A , α ). We can h ( a ))( ξ h ) for a ∈ A . g ∈ G a g u g as E ( au ∗ recover the coefficient a h of u h in a = � h ), and thus get We take C ∗ r ( G , A , α ), the reduced crossed product of the action g ∈ G a g u g for any a ∈ C ∗ a formal series � r ( G , A , α ). Unfortunately, this α : G → Aut( A ), to be the completion of A [ G ] in the norm � a � = � σ ( a ) � . series need not converge in any standard topology. (However, under It is a theorem that this norm does not depend on π as long as π is suitable conditions, its Ces` aro means converge to a in norm.) injective. N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 5 / 28 N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 6 / 28 Crossed products by Z Crossed products by homeomorphisms In this lecture, G = Z . An action α : Z → Aut( A ) is determined by its The irrational rotation algebra A θ , for θ ∈ R \ Q , is a famous example. By generator α 1 , which is an arbitrary automorphism of A ; we usually just call definition, it is the universal C*-algebra generated by unitaries u and v it α , and for α ∈ Aut( A ) we write C ∗ ( Z , A , α ) for the crossed product by satisfying vu = e 2 π i θ uv . We can rewrite the relation as uvu ∗ = e − 2 π i θ v . the action generated by α . We conventionally take u = u 1 the standard Take A = C ∗ ( v ) ∼ = C ( S 1 ), using the isomorphism sending v to the unitary in the crossed product corresponding to the generator 1 ∈ Z . function v ( ζ ) = ζ for ζ ∈ S 1 . Then u corresponds to the standard unitary (Note the change of notation: 1 is not the identity of Z .) Thus u n = u n . in C ∗ ( Z , C ( S 1 ) , α ) corresponding to generator 1 ∈ Z for α ∈ Aut( C ( S 1 )) We also let E : C ∗ ( Z , A , α ) → A be the standard conditional expectation determined by α ( v ) = e − 2 π i θ v . For general f ∈ C ( S 1 ), this is (picking out the coefficient of the group identity). We will usually have α ( f )( ζ ) = f ( e − 2 π i θ ζ ) for ζ ∈ S 1 . A = C ( X ); see below. For a compact Hausdorff space X and a homeomorphism h : X → X , we See the incomplete draft lecture notes on my website, Crossed Product use the automorphism α ( f )( x ) = f ( h − 1 ( x )) for f ∈ C ( X ) and x ∈ X to C*-Algebras and Minimal Dynamics , especially Sections 8 and 9, for a lot define an action of Z on X , and write C ∗ ( Z , X , h ) for C ∗ ( Z , C ( X ) , α ). For more on crossed products. Full and reduced crossed products exist in the irrational rotation algebra A θ , h ( ζ ) = e 2 π i θ ζ for ζ ∈ S 1 . much greater generality: A need not be unital, and G can be any locally compact group. N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 7 / 28 N. C. Phillips (U of Oregon) Large Subalgebras: Crossed Products by Z 5 June 2015 8 / 28