Ultraproducts in Functional Analysis Marius Junge Pisa, June 2008 - PowerPoint PPT Presentation

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space,

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space, in particular M n = B ( ℓ n 2 ).

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space, in particular M n = B ( ℓ n 2 ). Finite dimensional C ∗ -algebras are direct sums of matrix algebras.

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space, in particular M n = B ( ℓ n 2 ). Finite dimensional C ∗ -algebras are direct sums of matrix algebras. Every C ∗ -algebra is contained in some B ( H ).

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space, in particular M n = B ( ℓ n 2 ). Finite dimensional C ∗ -algebras are direct sums of matrix algebras. Every C ∗ -algebra is contained in some B ( H ). C ∗ ( F ∞ ), the universal algebra of infinitely many unitaries,

C ∗ -algebras A C ∗ -algebra is a Banach algebra with involution ∗ such that � x � 2 = � x ∗ x � . Examples: A = C ( K ), K compact. A = C 0 ( K ), K locally compact. B ( H ), the bounded operators on Hilbert space, in particular M n = B ( ℓ n 2 ). Finite dimensional C ∗ -algebras are direct sums of matrix algebras. Every C ∗ -algebra is contained in some B ( H ). C ∗ ( F ∞ ), the universal algebra of infinitely many unitaries, F ∞ free group in countably many generators.

Von Neumann algebras

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology:

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) .

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions,

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators.

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples:

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ).

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ),

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ), L ∞ (Ω , µ ; B ( H ))) (random matrices).

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ), L ∞ (Ω , µ ; B ( H ))) (random matrices). X ⊂ B ( H ) such that X ∗ ⊂ X ,

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ), L ∞ (Ω , µ ; B ( H ))) (random matrices). X ⊂ B ( H ) such that X ∗ ⊂ X , then X ′ = { T : Tx − xT = 0 , ∀ x ∈ X } is a vNa.

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ), L ∞ (Ω , µ ; B ( H ))) (random matrices). X ⊂ B ( H ) such that X ∗ ⊂ X , then X ′ = { T : Tx − xT = 0 , ∀ x ∈ X } is a vNa. Let G be a discrete group and λ ( g ) e h = e gh .

Von Neumann algebras A von Neumann algebra is a unital subalgebra of B ( H ) closed in the weak operator topology: T λ − → WOT T if ( h , T λ k ) − → λ ( h , Tk ) . Motivation: Functional calculus with measurable functions, spectral theory of unbounded operators. Examples: B ( H ). L ∞ (Ω , µ ), L ∞ (Ω , µ ; B ( H ))) (random matrices). X ⊂ B ( H ) such that X ∗ ⊂ X , then X ′ = { T : Tx − xT = 0 , ∀ x ∈ X } is a vNa. Let G be a discrete group and λ ( g ) e h = e gh . Then ′′ is a von Neumann algebra. VN ( G ) = λ ( G )

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace,

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace, i.e. a positive, normal functional with τ (1) = 1 and τ ( xy ) = τ ( yx ).

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace, i.e. a positive, normal functional with τ (1) = 1 and τ ( xy ) = τ ( yx ). Then ultraproduct N U ( N ω in vNa-lit) is the quotient of ℓ ∞ ( I , N ) with respect to i , U τ ( x ∗ I = { ( x i ) : lim i x i ) = 0 } .

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace, i.e. a positive, normal functional with τ (1) = 1 and τ ( xy ) = τ ( yx ). Then ultraproduct N U ( N ω in vNa-lit) is the quotient of ℓ ∞ ( I , N ) with respect to i , U τ ( x ∗ I = { ( x i ) : lim i x i ) = 0 } . Warning/Remark: 1) I is much larger than { ( x i ) : lim i � x i � = 0 } .

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace, i.e. a positive, normal functional with τ (1) = 1 and τ ( xy ) = τ ( yx ). Then ultraproduct N U ( N ω in vNa-lit) is the quotient of ℓ ∞ ( I , N ) with respect to i , U τ ( x ∗ I = { ( x i ) : lim i x i ) = 0 } . Warning/Remark: 1) I is much larger than { ( x i ) : lim i � x i � = 0 } . 2) However, ( N U ) ∗ is a two-sided ideal in � U N ∗ .

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Von Neumann algebra ultraprowers Let N be a von Neumann algebra and τ be a trace, i.e. a positive, normal functional with τ (1) = 1 and τ ( xy ) = τ ( yx ). Then ultraproduct N U ( N ω in vNa-lit) is the quotient of ℓ ∞ ( I , N ) with respect to i , U τ ( x ∗ I = { ( x i ) : lim i x i ) = 0 } . Warning/Remark: 1) I is much larger than { ( x i ) : lim i � x i � = 0 } . 2) However, ( N U ) ∗ is a two-sided ideal in � U N ∗ . 3) The Chang-Keisler theorem for ultraproducts in the vNa-sense is missing.

Property Γ

Property Γ N has property Γ if N ′ ∩ N U � = C . Example: 1) Let R = ⊗ n ∈ N M 2 the infinite tensor product of 2 × 2 matrices. Then R has property Γ.

Property Γ N has property Γ if N ′ ∩ N U � = C . Example: 1) Let R = ⊗ n ∈ N M 2 the infinite tensor product of 2 × 2 matrices. Then R has property Γ. Indeed, every von Neumann algebra which is the WOT closure of finite dimensional C ∗ -algebras, has this property (hyperfinite).

Property Γ N has property Γ if N ′ ∩ N U � = C . Example: 1) Let R = ⊗ n ∈ N M 2 the infinite tensor product of 2 × 2 matrices. Then R has property Γ. Indeed, every von Neumann algebra which is the WOT closure of finite dimensional C ∗ -algebras, has this property (hyperfinite). VN ( G ) is hyperfinite iff G is amenable.

Property Γ N has property Γ if N ′ ∩ N U � = C . Example: 1) Let R = ⊗ n ∈ N M 2 the infinite tensor product of 2 × 2 matrices. Then R has property Γ. Indeed, every von Neumann algebra which is the WOT closure of finite dimensional C ∗ -algebras, has this property (hyperfinite). VN ( G ) is hyperfinite iff G is amenable. 2) Let F n be the free group in n generators. Then VN ( F n ) does not have property Γ (Murray/von Neumann).

Property Γ N has property Γ if N ′ ∩ N U � = C . Example: 1) Let R = ⊗ n ∈ N M 2 the infinite tensor product of 2 × 2 matrices. Then R has property Γ. Indeed, every von Neumann algebra which is the WOT closure of finite dimensional C ∗ -algebras, has this property (hyperfinite). VN ( G ) is hyperfinite iff G is amenable. 2) Let F n be the free group in n generators. Then VN ( F n ) does not have property Γ (Murray/von Neumann). Hence, VN ( F n ) is not hyperfinite.

More recent results

More recent results Recently (03) Christensen, Smith, Sinclair and Pop showed that for factors with property Γ the bounded cohomology groups vanish.

More recent results Recently (03) Christensen, Smith, Sinclair and Pop showed that for factors with property Γ the bounded cohomology groups vanish. Popa showed that for Q ⊂ N , Q contains no hyperfinite summand if and only if Q ′ ∩ ( N ∗ N ) U ⊂ ( N ∗ 1) U holds for the free product.

More recent results Recently (03) Christensen, Smith, Sinclair and Pop showed that for factors with property Γ the bounded cohomology groups vanish. Popa showed that for Q ⊂ N , Q contains no hyperfinite summand if and only if Q ′ ∩ ( N ∗ N ) U ⊂ ( N ∗ 1) U holds for the free product. This can be used to show that for every sub von Neumann algebra Q of VN ( F n ) such that Q ′ ∩ L ( VN ( F n )) has no atoms, then Q is hyperfinite (due to Ozawa).

More recent results Recently (03) Christensen, Smith, Sinclair and Pop showed that for factors with property Γ the bounded cohomology groups vanish. Popa showed that for Q ⊂ N , Q contains no hyperfinite summand if and only if Q ′ ∩ ( N ∗ N ) U ⊂ ( N ∗ 1) U holds for the free product. This can be used to show that for every sub von Neumann algebra Q of VN ( F n ) such that Q ′ ∩ L ( VN ( F n )) has no atoms, then Q is hyperfinite (due to Ozawa). Popa has very successfully studied defomration/rigidity result in von Neumann algebras.

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Embedding in R U

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Embedding in R U Problem 1: Let be a von Neumann algebra N with a nice trace. Is there a trace preserving embedding of N in R U ?

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Embedding in R U Problem 1: Let be a von Neumann algebra N with a nice trace. Is there a trace preserving embedding of N in R U ? Remark: Then the range is automatically complemented with a conditional expectation E : R U → N , E ( axb ) = aE ( x ) b , a , b ∈ N , x ∈ R U .

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Embedding in R U Problem 1: Let be a von Neumann algebra N with a nice trace. Is there a trace preserving embedding of N in R U ? Remark: Then the range is automatically complemented with a conditional expectation E : R U → N , E ( axb ) = aE ( x ) b , a , b ∈ N , x ∈ R U . A good way to understand this is to ask wheather for a finite set x 1 , ..., x m ⊂ N there are matrices y 1 , ..., y m ∈ M n of n × n matrices such that | τ ( x i 1 · · · x i k ) − tr n ( y i 1 · · · y i k ) | < ε ?

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Kirchberg’s theorem

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Kirchberg’s theorem Problem 2: Let N be a arbitrary von Neumann algebra. Is there an isometric embedding of the predual N ∗ in � U B ( H ) ∗ ?

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Kirchberg’s theorem Problem 2: Let N be a arbitrary von Neumann algebra. Is there an isometric embedding of the predual N ∗ in � U B ( H ) ∗ ? Problem 3: Let N be an arbitrary von Neumann algebra. Is there U B ( H ) ∗ ) ∗ (or B ( H ) ∗∗ ) with a normal an embedding in ( � conditional expectation E : � U B ( H ) → N ?

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Kirchberg’s theorem Problem 2: Let N be a arbitrary von Neumann algebra. Is there an isometric embedding of the predual N ∗ in � U B ( H ) ∗ ? Problem 3: Let N be an arbitrary von Neumann algebra. Is there U B ( H ) ∗ ) ∗ (or B ( H ) ∗∗ ) with a normal an embedding in ( � conditional expectation E : � U B ( H ) → N ? Problem 4: Is there only one norm on C ∗ ( F ∞ ) ⊗ C ∗ ( F ∞ ) which makes the tensor product a C ∗ -algebra?

General remarks Local properties Von Neumann algebras Connes’ embedding problem Kirchberg’s theorem Ultraproduct techniqu Kirchberg’s theorem Problem 2: Let N be a arbitrary von Neumann algebra. Is there an isometric embedding of the predual N ∗ in � U B ( H ) ∗ ? Problem 3: Let N be an arbitrary von Neumann algebra. Is there U B ( H ) ∗ ) ∗ (or B ( H ) ∗∗ ) with a normal an embedding in ( � conditional expectation E : � U B ( H ) → N ? Problem 4: Is there only one norm on C ∗ ( F ∞ ) ⊗ C ∗ ( F ∞ ) which makes the tensor product a C ∗ -algebra? Theorem ( 94) The four problems are all equivalent.

L p spaces

L p spaces Let N be a von Neumann algebra with trace τ .

L p spaces Let N be a von Neumann algebra with trace τ . The L p spaces is defined by √ � x � p = [ τ ( | x | p )] 1 / p x ∗ x . , | x | =

L p spaces Let N be a von Neumann algebra with trace τ . The L p spaces is defined by √ � x � p = [ τ ( | x | p )] 1 / p x ∗ x . , | x | = Theorem (J. Parcet–NC Rosenthal theorem) Let X ⊂ L 1 ( N ) be a reflexive subspace, then X is isomorphic to subspace of L p ( N ) for some p > 1 .

L p spaces Let N be a von Neumann algebra with trace τ . The L p spaces is defined by √ � x � p = [ τ ( | x | p )] 1 / p x ∗ x . , | x | = Theorem (J. Parcet–NC Rosenthal theorem) Let X ⊂ L 1 ( N ) be a reflexive subspace, then X is isomorphic to subspace of L p ( N ) for some p > 1 .Indeed, there exists a positive d ∈ L 1 ( N ) and u : X → L p such that x = d 1 − 1 / p u ( x ) + u ( x ) d 1 − 1 / p .

L p spaces Let N be a von Neumann algebra with trace τ . The L p spaces is defined by √ � x � p = [ τ ( | x | p )] 1 / p x ∗ x . , | x | = Theorem (J. Parcet–NC Rosenthal theorem) Let X ⊂ L 1 ( N ) be a reflexive subspace, then X is isomorphic to subspace of L p ( N ) for some p > 1 .Indeed, there exists a positive d ∈ L 1 ( N ) and u : X → L p such that x = d 1 − 1 / p u ( x ) + u ( x ) d 1 − 1 / p . Remark: Many ultra product techniques in the proof

L p spaces Let N be a von Neumann algebra with trace τ . The L p spaces is defined by √ � x � p = [ τ ( | x | p )] 1 / p x ∗ x . , | x | = Theorem (J. Parcet–NC Rosenthal theorem) Let X ⊂ L 1 ( N ) be a reflexive subspace, then X is isomorphic to subspace of L p ( N ) for some p > 1 .Indeed, there exists a positive d ∈ L 1 ( N ) and u : X → L p such that x = d 1 − 1 / p u ( x ) + u ( x ) d 1 − 1 / p . Remark: Many ultra product techniques in the proof +results of Pisier.

Theorem (J-NC Fubini theorem) ( N i ) and ( M j ) be von Neumann algebras and k x k ( i ) ⊗ y k ( j ) a finite tensor. z = �