A word on duality Jonathan Turk Arizona State University October 21, 2020
Overview • Describe general duality theory • Present three basic duality theories • Discuss similar duality-type results in other areas of mathematics • Provide a categorical approach of duality
What is duality? Duality refers to a reverse process, i.e., a “dual” process. Idea: If object X is obtained through some construction, how much information about the original object can be recovered? What process will give us this information? Easy example: Given H ∗ for some Hilbert space H , we can recover H by forming the double dual H ∗∗ . Duality theories: • Pontryagin duality • Crossed product duality • Landstad duality
Pontryagin duality
Pontryagin duality G - locally compact abelian group � G - set of all continuous homomorphisms from G to T � G becomes a locally compact group, known as the dual group of G , with the following structure: Operation: pointwise multiplication Topology: uniform convergence on compact sets = � Pontryagin duality theorem: G ∼ � G s : � For s ∈ G , define � G → T by � s ( γ ) = γ ( s ). The map s �→ � s from G to � � G is an isomorphism.
Crossed product duality
W ∗ -crossed products Let G be a locally compact group, A a von Neumann algebra on H , and α an action of G on A by automorphisms. Then X := L 2 ( G , H ) is a Hilbert space with left Haar measure. α : A → B ( X ); α ( a ) f : x �→ α x − 1 f ( x ) Define ˜ ˜ λ ( y ) f : x �→ f ( y − 1 x ) λ : G → B ( X ); The W ∗ -crossed product A ⋊ α G is the von Neumann algebra on � � α ( a ) λ ( x ) : a ∈ A , x ∈ G X generated by the set ˜ . • The algebraic structure of A ⋊ α G does not depend on H . • A ⋊ α G does not satisfy a universal property.
C ∗ -crossed products If G is a locally compact group and A is a C ∗ -algebra, an action of G on A is a homomorphism α : G → Aut A such that for each a ∈ A , the map s �→ α s ( a ) is continuous. A dynamical system is a triple ( A , G , α ) where A is a C ∗ -algebra, G is a locally compact group, and α is an action of G on A . The (full) crossed product A ⋊ α G is the completion of the convolution *-algebra C c ( G , A ) with the universal norm. The reduced crossed product A ⋊ α, r G is the completion with reduced norm.
Crossed product duality Idea: “Undo” a crossed product by taking a crossed product. Note: Due to collapsing, we can’t recover the original algebra in general. Four crossed product duality theorems: • Takesaki (1973) • Takai (1975) • Imai-Takai (1978) • Katayama (1985)
Takesaki duality Suppose A is a von Neumann algebra on H , G is an abelian locally compact group, and α : G � A . � � Define µ : � L 2 ( G , H ) G → U µ ( p ) ξ : g �→ p ( g ) ξ ( g ) ; α : � G → Aut( A ⋊ α G ); α p : x �→ µ ( p ) x µ ( p ) ∗ Define � � � � � � α � G ∼ L 2 ( G ) Takesaki duality theorem: A ⋊ α G = A ⊗ B ⋊ � • ⊗ refers to the von Neumann tensor product � � L 2 ( G ) • B is a type I factor • If A is properly infinite and G is separable, then � � G ∼ α � A ⋊ α G ⋊ � = A
Application: Structure of von Neumann algebras In Takesaki’s original paper on duality, he proved that every type III von Neumann algebra may be expressed uniquely in the form A ⋊ α R where A is a type II ∞ von Neumann algebra and α leaves the trace relatively invariant. He also proved a structure theorem for hyperfinite II 1 factors.
Takai duality Let ( A , G , α ) be a dynamical system where G is abelian. For � � γ ∈ � G , define � α γ ∈ Aut C c ( G , A ) by � α γ ( f )( s ) = γ ( s ) f ( s ). � � Then � α γ ∈ Aut C c ( G , A ) , and hence extends to an element of � � Aut A ⋊ α G . � � α : � is an action of � G → Aut The map � A ⋊ α G G on A ⋊ α G ; this is called the dual action . The triple ( A ⋊ α G , � G , � α ) is a dynamical system, known as the dual system of ( A , G , α ). � � G ∼ α � L 2 ( G ) = A ⊗ K Takai duality theorem: ( A ⋊ α G ) ⋊ �
Nonabelian versions Recall: If ( A , G , α ) is a dynamical system with G abelian, then the α is an action of � dual action � G on A ⋊ α G . Observe: If G is nonabelian, there is no dual group, so no dual action. We get around this by using coactions.
What is a coaction? � � δ G : C ∗ ( G ) → M C ∗ ( G ) ⊗ C ∗ ( G ) Comultiplication: Satisfies δ G ( g ) = g ⊗ g for all g ∈ G (in some sense) (Full) coaction: Injective nondegenerate homomorphism � � A ⊗ C ∗ ( G ) δ : A → M � � (i) δ ( A ) 1 ⊗ C ∗ ( G ) ⊂ A ⊗ C ∗ ( G ) � � δ − − − − → A ⊗ C ∗ ( G ) A M (ii) � � id A ⊗ δ G δ � � � � δ ⊗ id G A ⊗ C ∗ ( G ) A ⊗ C ∗ ( G ) ⊗ C ∗ ( G ) M − − − − → M Using the reduced group C ∗ -algebra C ∗ r ( G ) instead gives a so-called reduced coaction
Covariant representations A representation of a dynamical system ( A , G , α ) on a Hilbert space H should encode information about A , G , and α . Such a representation is called a covariant representation . Rigorously, a covariant representation of ( A , G , α ) on H is a pair ( π, U ) where • π : A → B ( H ) is a representation (encodes A ) • U : G → U ( H ) is a representation (encodes G ) � � = U s π ( a ) U ∗ • π α s ( a ) ∀ a ∈ A (encodes α ) s Theorem (Raeburn): The crossed product A ⋊ α G is universal for covariant representations of ( A , G , α ) (in the appropriate sense)
Crossed products by coactions � � If δ : A → M A ⊗ C ∗ ( G ) is a coaction, a covariant representation of ( A , G , δ ) is defined analogously to those of dynamical systems. The crossed product A ⋊ δ G is defined as the universal C ∗ -algebra for covariant representations of ( A , G , δ ) (in the appropriate sense).
Imai-Takai duality Idea: “Undo” a classical crossed product by taking a coaction crossed product. Let ( A , G , α ) be a dynamical system. α of � Recall: For Takai duality, we use α to produce an action � G on A ⋊ α G . For Imai-Takai duality, we use α to create a coaction � α of G on A ⋊ α G , called the dual coaction . � � � � α G ∼ L 2 ( G ) Imai-Takai duality theorem: = A ⊗ K A ⋊ α, r G ⋊ �
Katayama duality Idea: “Undo” a coaction crossed product by taking a classical crossed product. Let ( A , G , δ ) be a coaction. Recall: For Imai-Takai duality, we use an action to produce a coaction. For Katayama duality, we use the coaction δ to produce an action � δ of G on A ⋊ δ G . � � � � δ, r G ∼ L 2 ( G ) Katayama duality theorem: A ⋊ δ G = A ⊗ K . ⋊ �
Landstad duality
Landstad duality Idea: Recover a crossed product up to isomorphism as opposed to Morita equivalence. In addition, this duality theory characterizes crossed products by a locally compact group G . Note: Although we are still looking at crossed products, Landstad duality is not a type of crossed product duality. In his 1979 paper, Landstad originally presented four variations of his duality theorem: Reduced C ∗ -crossed products by an arbitrary group • • Full C ∗ -crossed products by an abelian group • von Neumann crossed products by an arbitrary group • von Neumann crossed products by an abelian group
Reduced C ∗ -crossed products by an arbitrary group Suppose B is a C ∗ -algebra and G is a locally compact group. Characterization: There is a C ∗ -algebra A and an action α : G � A such that B ∼ = A ⋊ α, r G if and only if there is a strictly continuous homomorphism λ : G → UM ( B ) and a reduced � � coaction δ of G on B such that δ λ ( x ) = λ ( x ) ⊗ x ∀ x ∈ G . Recovery: Given the homomorphism λ and the coaction δ , we can compute A and α up to isomorphism. Let A be the set of elements a ∈ M ( B ) which satisfy the following: • δ ( a ) = a ⊗ I • a λ ( f ) , λ ( f ) a ∈ B for all f ∈ C c ( G ). x �→ λ ( x ) a λ ( x − 1 ) is norm-continuous. • Define α : G → Aut A by α x : a �→ λ ( x ) a λ ( x − 1 ).
Full C ∗ -crossed products by an abelian group Let B be a C ∗ -algebra and G an abelian locally compact group. Characterization: There is a C ∗ -algebra A and an action α : G � A such that B ∼ = A ⋊ α G if and only if there is a continuous homomorphism λ : G → UM ( B ) and an action � � α : � = γ ( x ) λ ( x ) ∀ x ∈ G , γ ∈ � � G � B such that � α γ λ ( x ) G . Recovery: Given the homomorphism λ and the action � α , we can compute A and α up to isomorphism. Let A be the set of elements a ∈ M ( B ) which satisfy the following: α γ ( a ) = a ∀ γ ∈ � • � G • a λ ( f ) , λ ( f ) a ∈ B for all f ∈ C c ( G ). x �→ λ ( x ) a λ ( x − 1 ) is norm-continuous. • Define α : G → Aut A by α x : a �→ λ ( x ) a λ ( x − 1 ).
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