A word on duality Jonathan Turk Arizona State University October - - PowerPoint PPT Presentation
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
Jonathan Turk Arizona State University October 21, 2020
mathematics
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:
G - locally compact abelian group
G, with the following structure: Operation: pointwise multiplication Topology: uniform convergence on compact sets Pontryagin duality theorem: G ∼ =
For s ∈ G, define s : G → T by s(γ) = γ(s). The map s → s from G to
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 := L2(G, H) is a Hilbert space with left Haar measure. Define ˜ α : A → B(X); ˜ α(a)f : x → αx−1f (x) λ : G → B(X); λ(y)f : x → f (y−1x) The W ∗-crossed product A ⋊α G is the von Neumann algebra on X generated by the set
α(a)λ(x) : a ∈ A, x ∈ G
If G is a locally compact group and A is a C ∗-algebra, an action of G on A is a homomorphism α : G → AutA such that for each a ∈ A, the map s → αs(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 Cc(G, A) with the universal norm. The reduced crossed product A ⋊α,r G is the completion with reduced norm.
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:
Suppose A is a von Neumann algebra on H, G is an abelian locally compact group, and α : G A. Define µ : G → U
µ(p)ξ : g → p(g)ξ(g) Define α : G → Aut(A ⋊α G);
Takesaki duality theorem:
α
G ∼ = A ⊗ B
α
G ∼ = A
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 II1 factors.
Let (A, G, α) be a dynamical system where G is abelian. For γ ∈ G, define αγ ∈ Aut
αγ(f )(s) = γ(s)f (s). Then αγ ∈ Aut
Aut
The map α : G → Aut
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). Takai duality theorem: (A ⋊α G) ⋊
α
G ∼ = A ⊗ K
Recall: If (A, G, α) is a dynamical system with G abelian, then the dual action α is an action of 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.
Comultiplication: δG : C ∗(G) → M
(Full) coaction: Injective nondegenerate homomorphism δ : A → M
δ(A)
(ii) A
δ
− − − − → M
idA⊗δG M
− − − − → M
r (G) instead gives a
so-called reduced coaction
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
s
∀a ∈ A (encodes α) Theorem (Raeburn): The crossed product A ⋊α G is universal for covariant representations of (A, G, α) (in the appropriate sense)
If δ : A → M
The crossed product A ⋊δ G is defined as the universal C ∗-algebra for covariant representations of (A, G, δ) (in the appropriate sense).
Idea: “Undo” a classical crossed product by taking a coaction crossed product. Let (A, G, α) be a dynamical system. Recall: For Takai duality, we use α to produce an action α of 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. Imai-Takai duality theorem:
α G ∼
= A ⊗ K
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
Katayama duality theorem:
δ,r G ∼
= A ⊗ K
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:
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
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:
Define α : G → AutA by αx : a → λ(x)aλ(x−1).
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
G B such that αγ
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:
G
Define α : G → AutA by αx : a → λ(x)aλ(x−1).
A Busby-Smith dynamical system is basically a dynamical system where the action is “twisted” by a 2-cocycle. Such systems are of the form (A, G, α, u) where u is a 2-cocycle (the “twist”), i.e, satisfies the cocycle identity: αr
∀r, s, t ∈ G The “twisted crossed product” A ⋊α,u G is formed (somewhat) similarly to the W ∗-crossed product. There is a duality result (Quigg 1986) for Busby-Smith dynamical
α of G on A ⋊α,u G such that
α G ∼
= A ⊗ K
A representation (H, π) of A induces a representation
π, ρπ)
The induced representation induces a C ∗-subalgebra C ∗(indπ) of B
Then G coacts on C ∗(indπ) by some coaction δπ, so we can take the crossed product of the system
The isomorphism below generalizes the duality theorem previously stated (Quigg 1986): C ∗(indπ) ⋊δπ G ∼ = ˜ π(A) ⊗ K
In some cases, we get a decomposition theorem of A ⋊α,u G. (Packer and Raeburn 1989) If N is a closed normal subgroup of G, then there is a twisted action (β, v) of G/N on A ⋊α,u N such that
= A ⋊α,u G.
Suppose G is a group and E is a directed graph. An action of G on E is a homomorphism γ : G → AutE. γ is free if for all g ∈ G and all v ∈ E 0, γgv = v implies g = 1G. Taking the quotient of E by the orbits of G gives a directed graph E/G.
Suppose E and F are directed graphs. The cartesian product E × F becomes a directed graph in a natural way:
0 := E 0 × F 0
1 := E 1 × F 1 r(e, f ) :=
directed graph, the skew-product E ×c G:
0 := E 0 × G
1 := E 1 × G r(e, g) :=
Skew products are closely related to crossed products. (Kumjian and Pask 1997) If G is an abelian group, E is a directed graph in which every vertex emits an edge, and c : E 1 → G, then there is an action α of G on C ∗(E) such that C ∗(E ×c G) ∼ = C ∗(E) ⋊α G This result and Takai duality may be used to show that C ∗(E) is in the bootstrap class N, i.e., the UCT applies to C ∗(E).
(Kumjian and Pask 1997) Suppose E is a directed graph in which every vertex emits an edge. Let Z be the directed graph · · · − → • − → • − → • − → • − → · · · Define c : E 1 → Z by c(e) = 1 for all e ∈ E 1. Then E ×c Z = E × Z. The graph E × Z has no loops, so C ∗(E × Z), and hence C ∗(E) ⋊α T, is an AF algebra. Takai duality: C ∗(E) ⊗ K
∼ =
α Z.
It follows by properties of N that C ∗(E) is in N.
Suppose G is a countable group, E is a locally finite directed graph in which every vertex emits an edge, and c is a labelling. Given C ∗(E ×c G), we can recover C ∗(E) up to stable isomorphism. There is a natural free action γ of G on the skew-product E ×c G by left-multiplication. γ0
g(e, h) = (e, gh)
γ1
g(v, h) = (v, gh)
Then γ induces an action γ (abusing notation) of G on C ∗(E ×c G). Taking the crossed product gives us C ∗(E ×c G) ⋊γ G ∼ = C ∗(E) ⊗ K
Now suppose G is a countable group, E is a locally finite directed graph, and α : G → AutE is a free action. Let α be the induced action of G on C ∗(E). Given C ∗(E) and α, we can recover the quotient graph C ∗-algebra C ∗ E/G
C ∗(E) ⋊α G ∼ = C ∗ E/G
Previously, we saw that for abelian groups, C ∗(E ×c G) may be recognized as a crossed product of C ∗(E) by the dual group G. It’s natural to ask if similar results can be obtained for nonabelian
It says that if E is a row-finite directed graph, G is a discrete group, and c : E 1 → G, then there is a coaction δ of G on C ∗(E) such that C ∗(E ×c G) ∼ = C ∗(E) ⋊δ G. (Kaliszewski, Quigg, Raeburn 1999)
Recall: G naturally induces an action γ on C ∗(E ×c G). It turns out that C ∗(E ×c G) ⋊γ,r G ∼ =
δ,r G.
Given C ∗(E ×c G) and γ, we can use this result along with Katayama duality to recover C ∗(E) up to Morita equivalence: C ∗(E ×c G) ⋊γ,r G ∼ =
δ,r G ∼
= C ∗(E) ⊗ K
(Kaliszewski, Quigg, Raeburn 1999)
Idea: If F : C → D is a functor between categories, we want to find a functor from D to C which undoes F up to isomorphism. Basic examples Gelfand duality: Suppose C is the category of LCH spaces and D is the category of commutative C ∗-algebras. Define F : C → D by FX = C0(X) and G : D → C by GA = Ω(A). Pontryagin duality: Suppose C is the category of abelian locally compact groups. Define F : C → C by FG =
F undoes itself up to isomorphism. A more difficult example is Landstad duality.
Suppose G is a locally compact group. We create the category C(G): Objects: Ordered pairs (A, α) where A is a C ∗-algebra and α is an action of G on A Morphisms: A morphism φ : (A, α) → (B, β) is a homomorphism φ : A → B which is α − β equivariant.
Now we create the category D(G): Objects: Ordered 3-tuples (B, δ, λ) where λ : G → UM(B) is a strictly continuous homomorphism, δ is a reduced coaction of G on B, and δ
Morphisms: A morphism ϕ : (B, δ, λ) → (C, ε, γ) is a δ − ε equivariant homomorphism ϕ : B → C such that ϕ ◦ λ = γ.
Recall: If (A, α) ∈ C(G), then (the forward direction of) Landstad’s duality theorem tells us that there is a reduced coaction δ of G on A ⋊α,r G and a strictly continuous homomorphism λ : G → UM
x ∈ G. Recall: Given (A, α) ∈ C(G), there’s a dual coaction α of G on A ⋊α,r G. There’s also a universal strictly continuous homomorphism iG : G → UM(A ⋊α,r G) such that α
x ∈ G. We define the functor F : C(G) → D(G) by F(A, α) =
α, iG
Recall: If (B, δ, λ) ∈ D(G), Landstad duality tells us how to find an action (A, α) such that B ∼ = A ⋊α,r G. We define the functor G : D(G) → C(G) by G(B, δ, λ) = (A, α), where A and α are computed as in Landstad’s duality theorem. The functor G undoes F up to isomorphism.
Alex Kumjian and David Pask. “C∗-Algebras of Directed Graphs and Group Actions.” Ergodic theory and dynamical systems 19.6 (1999): 1503–1519. Web. Hiroshi Takai. “On a Duality for Crossed Products of C∗-Algebras.” Journal of functional analysis 19.1 (1975): 25–39. Web. John C. Quigg. “Duality for Reduced Twisted Crossed Products of C∗–Algebras.” Indiana University mathematics journal 35.3 (1986): 549–571. Print. Judith A. Packer and Iain Raeburn. “Twisted Crossed Products of C∗-Algebras.” Mathematical proceedings of the Cambridge Philosophical Society 106.2 (1989): 293–311. Web. Magnus B. Landstad. “Duality Theory for Covariant Systems.” Transactions of the American Mathematical Society 248.2 (1979): 223–267. Web.
(1973) 249-310.
COACTIONS.” Journal of operator theory 46.2 (2001): 411–433. Print. Sho Imai and Hiroshi Takai. “On a duality for C∗-crossed products by a locally compact group.” J. Math. Soc. Japan 30 (1978), no. 3, 495-504. Yoshikazu Katayama. “TAKESAKI’S DUALITY FOR A NON-DEGENERATE CO-ACTION.” Mathematica scandinavica 55.1 (1984): 141–151. Web.