Mathematics in the Hyperfinite World Evgeny Gordon Mathematics and - PowerPoint PPT Presentation

Mathematics in the Hyperfinite World Evgeny Gordon Mathematics and Computer Science Department Eastern Illinois University May, 2006

Harmonic analysis on finite abelian groups ◮ G a finite abelian group ◮ Dual group � G = Hom( G , S 1 ) ◮ S 1 = { z ∈ C | | z | = 1 } ◮ Pontrjagin Duality: ◮ G ≃ � � G G → S 1 where κ g ( χ ) = χ ( g ) → κ g : � ◮ g �− ◮ The Haar integral I ( f ) = ∆ � f ( g ). g ∈ G ◮ The Fourier transform: F ∆ : C G → C � G ◮ F ∆ ( f )( χ ) = ∆ � f ( g ) χ ( g ) , g ∈ G � ϕ ( χ ) χ ( g ) . ◮ F − 1 1 ∆ ( ϕ )( g ) = | G | ∆

Harmonic analysis on the nonstandard hulls of hyperfinite abelian groups ◮ G - a hyperfinite abelian group; ◮ G b ⊆ G a σ -subgroup; ◮ G 0 ⊆ G b a π -subgroup. ◮ Topology on G # = G b / G 0 ◮ For A ⊆ G 0 put i ( A ) = { a ∈ A | a + G 0 ⊆ A } . ◮ T = { i ( F ) # | G 0 ⊆ F ⊆ G b and F is internal } . - a base of neighborhoods of zero. ◮ Proposition The topology T is locally compact iff for any internal set F ⊃ G 0 and for any internal set B ⊆ G b there exists standardly finite set K ⊆ B such that B ⊆ K + F. ◮ Corollary 1). For every internal set F ⊆ G b the set F # is compact. 2). Every compact set K ⊆ G # is contained in some such F # .

◮ Corollary K ⊆ G # is a compact open subgroup iff K = H # , where H ⊃ G 0 is an internal subgroup of G b . ◮ If a locally compact group H is topologically isomorphic to G # , then we say that the triple ( G , G b , G 0 ) represents the H ◮ C 0 ( G # ) the set of all continuous functions with compact support on G # ◮ C 0 ( G ) the set of all internal S -continuous functions, whose support is contained in G b . ◮ Proposition A function f ∈ C 0 ( G # ) iff there exists an internal function ϕ ∈ C 0 ( G ) such that supp ϕ ⊆ G b and and for every g ∈ G b holds f ( g # ) = ◦ ϕ ( g ) . In this case we denote f by ϕ # .

Haar integral on G # ◮ A positive hyperreal number ∆ is a normalizing multiplier (n.m.) if for every internal set F , G 0 ⊆ F ⊆ G b , holds ◦ (∆ · | F | ) < + ∞ . ◮ If ∆ is an n.m., then a hyperreal number ∆ 1 is an n.m. iff 0 < ◦ � � ∆ 1 < + ∞ . ∆ ◮ Theorem If ∆ is an n.m., then the functional I on C 0 ( G # ) defined for every ϕ ∈ C 0 ( G ) by the formula I ( ϕ # ) = ◦ I ∆ ( f ) , is the Haar integral on G # .

Dual group � G # ◮ � G – (internal) group dual to G ; ◮ � G b = { χ ∈ � G | χ ↾ G 0 ≈ 1 } ; ◮ � G 0 = { χ ∈ � G | χ ↾ G b ≈ 1 } ; G # = � ◮ � G b / � G 0 . ◮ α # ∈ � G # �− → ψ ( α # ) ∈ � G # , α ∈ � G b ; ◮ ψ ( α # )( g # ) = ◦ α ( g ). ◮ Proposition G # → ψ ( � G # ) ⊆ � The mapping ψ : � G # is a topological isomorphism.

Theorem 1). Suppose that there exits an internal subgroup K ⊆ G b , G 0 ⊆ K. Then the following statements hold. G # ) = � G # is canonically isomorphic to � a). ψ ( � G # , thus � G # . D = ( | G | ∆) − 1 is a normalizing b). The hyperreal number � multiplier for � G c). Let f ∈ L 1 ( G # ) and ϕ be an S-integrable lifting of f . Then the Fourier transform on G F ∆ ( ϕ ) is an S-continuous G and the linear operator F : L 1 ( G # ) → C ( � function on � G # ) defined by the formula F ( f ) = F ∆ ( ϕ ) # is the Fourier transform on G # . The operator defined in the is the inverse Fourier transform on � similar way by F − 1 G # . ∆

◮ Theorem For every locally compact group H there exists a triple ( G , G b , G 0 ) representing H that satisfies the statements a) – c) of the first part of the theorem. ◮ Definition We say that a hyperfinite group G approximates a locally compact group H if there exist an internal injective map j : G → ∗ H that satisfies the following conditions: 1. ∀ h ∈ H ∃ g ∈ G ( j ( g ) ≈ h ) ; 2. ∀ g 1 , g 2 ∈ j − 1 ( ns ( ∗ H )) ( j ( g 1 ± g 2 ) ≈ j ( g 1 ) ± j ( g 2 )) . In this case we say that the pair ( G , j ) is a hyperfinite approximation of H. ◮ ( G , j ) �− → ( G , G b , G 0 ); ◮ G b = { g ∈ G | j ( g ) ∈ ns( H ) } , G 0 = { g ∈ G | j ( g ) ≈ 0 } .

Hyperfinite representations of locally compact non-commutative groups ◮ G – a non-commutative hyperfinite group. ◮ G b – a σ -subgroup, G 0 ⊆ G b – a π -subgroup, which is normal in G b . ◮ G # = G b / G 0 . ◮ For A ⊆ G put i ( A ) = { a ∈ G aG 0 ⊆ A } . ◮ T = { i ( F ) # | G 0 ⊆ F ⊆ G b and F is internal } form a base of a topology on G # . ◮ Proposition The topology T is locally compact iff for any internal set F ⊃ G 0 and for any internal set B ⊆ G b there exists standardly finite set K ⊆ B such that B ⊆ K · F. ◮ Corollary 1). For every internal set F ⊆ G b the set F # is compact. 2). Every compact set K ⊆ G # is contained in some such F # .

◮ Theorem If ∆ is a normalizing multiplier, then the positive functional I on C 0 ( G # ) defined by the formula I ( f # ) = ◦ (∆ � f ( g )) is left and g ∈ G right Haar integral. ◮ Corollary The group G # is unimodular. ◮ Definition A locally compact group H is weakly approximable by finite groups if there exists a triple ( G , G b , G 0 ) representing H. The group H is strongly approximable by finite groups if has a hyperfinite approximation . ◮ Theorem A compact Lie group H is strongly approximable by finite groups iff it has arbitrary dense finite subgroups.

Definition We say that a groupoid ( Q , ◦ ) is a quasigroup if for an arbitrary a , b ∈ Q each of the equations a ◦ x = b and x ◦ a = b has a unique solution. If it holds only for the first (second) equation, then we say that ( Q , ◦ ) is a left (right) quasigroup. ◮ ( Q , ◦ ) a hyperfinite groupoid, ◮ Q b ⊆ Q a σ -subgroupoid, ◮ ρ a π -equivalence relation on Q , that is a congruence relation on Q b . ◮ For A ⊆ Q b put i ( A ) = { q ∈ Q b | ρ ( q ) ⊆ A } . Theorem If Q is a left quasigroup and ∆ is a normalizing multiplier, then the positive functional I on C 0 ( Q # ) defined by the formula   �  ∆  I ( f # ) = ◦ f ( q ) q ∈ Q is left invariant. If Q is a quasigroup, then I ( f ) is right invariant also.

Theorem 1) Every locally compact group is strongly approximable by finite left quasigroups. 2) A locally compact group is unimodular iff it is strongly approximable by finite quasigroups

Discrete groups ◮ The topology on Q # is discrete iff ρ is the equality relation. ◮ A discrete group G is weakly approximable by a hyperfinite groupoid Q if it is isomorphic to a σ -subgroupoid of Q . ◮ The group G is strongly approximable by the hyperfinite groupoid Q iff there exists an internal injective map j : Q → ∗ G such that j ↾ j − 1 ( G ) is a homomorphism. Theorem A discrete group G is amenable iff there exists a hyperfinite set H, G ⊆ H ⊆ ∗ G, and a binary operation ◦ : H × H → H that satisfy the following conditions: 1. ( H , ◦ ) is a left quasigroup; 2. G is a subgroup of the left quasigroup ( H , ◦ ) , i.e. ∀ a , b ∈ G a · b = a ◦ b. 3. ∀ a ∈ G |{ h ∈ H | a · h = a ◦ h }| ≈ 1 | H | .

Definition A discrete group G is sofic iff there exists a hyperfinite set H, G ⊆ H, and a binary operation ◦ : H × H → H that satisfy the following conditions: 1. ( H , ◦ ) is a left quasigroup; 2. G is a subgroup of the left quasigroup ( H , ◦ ) , i.e. ∀ a , b ∈ G a · b = a ◦ b. 3. ∀ a , b ∈ G |{ h ∈ H | ( a · b ) ◦ h = a ◦ ( b · h }| ≈ 1 | H | .

Theorem (Elek, Szabo) Let N be an infinite hyperreal number and S N an internal group of permutations of the set { 1 , . . . , N } . Consider its π normal subgroup = { α ∈ S N | { n ≤ N | α ( n ) = n }| S (0) ≈ 1 } . N N Then S ( N ) = S N / S (0) is a simple sofic group. Moreover, a group N G is sofic iff it is isomorphic to a subgroup of the group S ( N ) for some infinite N.

Hyperfinite representations of topological universal algebras ◮ θ a finite signature that contains only functional symbols, ◮ A = � A , θ � a hyperfinite algebra of the signature θ . ◮ A b = � A b , θ � - σ -subalgebra of A ◮ ρ a π -equivalence relation on A , that is a congruence relation on A b . ◮ a , b ∈ A : α ≈ β ⇋ � a , b � ∈ ρ . ◮ ϕ ( x 1 , . . . , x n ) a first order formula of the signature θ . ◮ ϕ ≈ the formula obtained from ϕ by replacing of every subformula t 1 = t 2 by the formula t 1 ≈ t 2 , t 1 , t 2 are θ -terms. Proposition For every a 1 , . . . a n ∈ A b A # | = ϕ ( a # 1 , . . . , a # n ) ⇐ ⇒ A b | = ϕ ≈ ( a 1 , . . . a n ) .

Hyperfinite representations of reals ◮ The floating point representation of reals: α = ± 10 p × 0 . a 1 a 2 . . . , (1) p ∈ Z , 0 ≤ a n ≤ 9, a 1 � = 0. ◮ P , Q hypernatural numbers; ◮ A PQ the hyperfinite set of all reals of the form (1), where | p | ≤ P and the mantissa contains no more than Q decimal digits. ◮ ⊕ , ⊗ binary operations on A PQ , ∗ stands for either + or × ◮ α, β ∈ A PQ : α ∗ β = ± 10 r × 0 . c 1 c 2 . . . .

 ± 10 r × 0 . c 1 c 2 . . . c Q if | r | ≤ P ,    ± 10 P × 0 . 99 . . . 9 if r > P , � �� � α ⊛ β =   Q digits  0 if r < − P . ◮ A PQ the algebra � A PQ , ⊕ , ⊗� ◮ ( A PQ ) b consists of all finite hyperreal numbers from A PQ ◮ ρ a restriction of the relation ≈ on R to A PQ . ◮ Then A # PQ ≃ R .