von Neumann Algebra 2 Note Generation Theorem

von Neumann Algebra 2

Note

Generation Theorem  

TANAKA Akio

[Main Theorem]

<Generation theorem>

Commutative von Neumann Algebra N is generated by only one self-adjoint operator.

[Proof outline]

N is generated by countable {A_n}.

A_n = *A_n

Spectrum deconstruction       A_n = ∫¹_-1  λdE_λ⁽ⁿ⁾

C*algebra that is generated by set { E_λ⁽ⁿ⁾ ; λ∈Q∩[-1, 1], n∈N}     A

A’’ = N

A is commutative.

I∈A

Existence of compact Hausdorff space Ω = Sp(A  )

A   = C(Ω)

Element corresponded with f∈C(Ω)     A∈A

N is generated by A.

[Index of Terms]

|A|Ⅲ7-5

|| . ||Ⅱ2-2

||x||Ⅱ2-2

<x, y>Ⅱ2-1

*algebraⅡ3-4

*homomorphismⅡ3-4

*isomorphismⅡ3-4

*subalgebraⅡ3-4

adjoint spaceⅠ12

algebraⅠ8

axiom of infinityⅠ1-8

axiom of power setⅠ1-4

axiom of regularityⅠ1-10

axiom of separationⅠ1-6

axiom of sumⅠ1-5

B ( H )Ⅱ3-3

Banach algebraⅡ2-6

Banach spaceⅡ2-3

Banach* algebraⅡ2-6

Banach-Alaoglu theoremⅡ5

basis of neighbor hoodsⅠ4

bicommutantⅡ6-2

bijectiveⅡ7-1

binary relationⅡ7-2

boundedⅡ3-3

bounded linear operatorⅡ3-3

bounded linear operator, B ( H )Ⅱ3-3

C* algebraⅡ2-8

cardinal numberⅡ7-3

cardinality, |A|Ⅱ7-5

characterⅡ3-6

character space (spectrum space), Sp( )Ⅱ3-6

closed setⅠ2-2

commutantⅡ6-2

compactⅠ3-2

complementⅠ1-3

completeⅡ2-3

countable setⅡ7-6

countable infinite setⅡ7-6

coveringⅠ3-1

commutantⅡ6-2

D ( )Ⅱ3-2

denseⅠ9

dom( )Ⅱ3-2

domain, D ( ), dom( )Ⅱ3-2

empty setⅠ1-9

equal distance operatorⅡ4-1

equipotentⅢ7-1

faithfulⅡ3-4

Gerfand representationⅡ3-7

Gerfand-Naimark theoremⅡ4

HⅡ3-1

Hausdorff spaceⅠ5

Hilbert spaceⅡ3-1

homomorphismⅡ3-4

idempotent elementⅡ9-1

identity elementⅡ9-1

identity operatorⅡ6-1

injectiveⅢ7-1

inner productⅡ2-1

inner spaceⅠ6

involution*Ⅰ10

linear functionalⅡ5-2

linear operatorⅡ3-2

linear spaceⅠ6

linear topological spaceⅠ11

locally compactⅠ3-2

locally vertexⅠ11

NⅢ3-8

N₁Ⅲ3-8

neighborhoodⅠ4

normⅡ2-2

normⅡ3-3

norm algebraⅡ5

norm spaceⅡ2-2

normalⅡ2-4

normalⅡ3-4

open coveringⅠ3-2

open setⅠ2-2

operatorⅡ3-2

ordinal numberⅡ7-3

productⅠ8

product setⅡ7-2

r( )Ⅱ2

R ( )Ⅱ3-2

ran( )Ⅱ3-2

range, R ( ), ran( )Ⅱ3-2

reflectiveⅠ12

relationⅢ7-2

representationⅡ3-5

ringⅠ7

Schwarz’s inequalityⅡ2-2

self-adjointⅡ3-4

separableⅡ7-7

setⅠ7

spectrum radius r( )Ⅱ2

Stone-Weierstrass theoremⅡ1

subalgebraⅠ8

subcoveringⅠ3-1

subringⅠ7

subsetⅠ1-3

subspaceⅠ2-3

subtopological spaceⅠ2-3

surjectiveⅢ7-1

system of neighborhoodsⅠ4

τ^s topologyⅡ7-9

τ^w topologyⅡ7-9

the second adjoint spaceⅠ12

topological spaceⅠ2-2

topologyⅠ2-1

total order in strict senseⅡ7-3

ultra-weak topologyⅢ6-4

unit sphereⅡ5-1

unitaryⅡ3-4

vertex setⅡ3-3

von Neumann algebraⅡ6-3

weak topologyⅡ5-3

weak * topologyⅡ5-3

zero elementⅡ9-1

[Explanation of indispensable theorems for main theorem]

ⅠPreparation

<0 Formula>

0-1 Quantifier

(i) Logic quantifier  ┐ ⋀  ⋁  → ∀ ∃

(ii) Equality quantifier  =

(iii) Variant term quantifier

(iiii) Bracket  [  ]

(v) Constant term quantifier

(vi) Functional quantifier

(vii) Predicate quantifier

(viii) Bracket  (   )

(viiii) Comma  ,

0-2 Term defined by induction

0-3 Formula defined by induction 

<1 Set>

1-1 Axiom of extensionality     ∀x∀y[∀z∈x↔z∈y]→x=y.

1-2 Set     a, b

1-3 a is subset of b.    ∀x[x∈a→x∈b].Notation is a⊂b. b-a = {x∈b ; x∉a} is complement of a.

1-4 Axiom of power set     ∀x∃y∀z[z∈y↔z⊂x]. Notation is P (a).

1-5 Axiom of sum     ∀x∃y∀z[z∈y↔∃w[z∈w∧w∈x]]. Notation is ∪a.

1-6 Axiom of separation     x, t= (t1, …, tn), formula φ(x, t)     ∀x∀t∃y∀z[z∈y↔z∈x∧φ(x, t)].

1-7 Proposition of intersection     {x∈a ; x∈b} = {x∈b; x∈a} is set by axiom of separation. Notation is a∩b.

1-8 Axiom of infinity     ∃x[0∈x∧∀y[y∈x→y∪{y}∈x]].

1-9 Proposition of empty set     Existence of set a is permitted by axiom of infinity. {x∈a; x≠x} is set and has not element. Notation of empty set is 0 or Ø.

1-10 Axiom of regularity     ∀x[x≠0→∃y[y∈x∧y∩x=0].

<2 Topology>

2-1

Set     X

Subset of power set P(X)     T

T that satisfies next conditions is called topology.

(i) Family of X’s subset that is not empty set     <A_i; i∈I>, A_i∈T→∪_i∈I Ai is belonged to T.       

(ii) A, B ∈T→ A∩B∈T

(iii) Ø∈T, X∈T.

2-2

Set having T, (X, T), is called topological space, abbreviated to X, being logically not confused.

Element of T is called open set.

Complement of Element of T is called closed set.

2-3

Topological space     (X, T)

Subset of X     Y

S ={A∩Y ; A∈T}

Subtopological space     (Y, S)   

Topological space is abbreviated to subspace.

<3 Compact>

3-1

Set     X

Subset of X     Y

Family of X’s subset that is not empty set     U = <U_i; i∈I>

U is covering of Y.     ∪U = ∪_i∈I ⊃Y

Subfamily of U 　　V = <U_i; i∈J > (J⊂I)

V is subcovering of U.

3-2

Topological space     X

Elements of U     Open set of X

U is called open covering of Y.

When finite subcovering is selected from arbitrary open covering of X, X is called compact.

When topological space has neighborhood that is compact at arbitrary point, it is called locally compact.

<4 Neighborhood>

Topological space     X

Point of X     a

Subset of X     A

Open set    B

a∈B⊂A

A is called neighborhood of a.

All of point a’s neighborhoods is called system of neighborhoods.

System of neighborhoods of point a     V(a)

Subset of V(a)     U

Element of U     B

Arbitrary element of V(a)     A

When B⊂A, U is called basis of neighborhoods of point a.

<5 Hausdorff space>

Topological space X that satisfies next condition is called Hausdorff space.

Distinct points of X     a, b        

Neighborhood of a     U

Neighborhood of b     V

U∩V = Ø

<6 Linear space>

Compact Hausdorff space     Ω

Linear space that is consisted of all complex valued continuous functions over Ω     C(Ω)

When Ω is locally compact, all complex valued continuous functions over Ω, that is 0 at infinite point is expressed by C₀(Ω).

<7 Ring>

Set     R

When R is module on addition and has associative law and distributive law on product, R is called ring.

When ring in which subset S is not φ satisfies next condition, S is called subring.

a, b∈S

ab∈S

<8 Algebra>

C(Ω) and C₀(Ω) satisfy the condition of algebra at product between points.

Subspace     A ⊂C(Ω) or A ⊂C₀(Ω)

When A is subring, A is called subalgebra.

<9 Dense>

Topological space     X

Subset of X     Y

Arbitrary open set that is not Ø in X     A

When A∩Y≠Ø, Y is dense in X.

<10 Involution>

Involution * over algebra A over C is map * that satisfies next condition.

Map * : A∈A ↦ A*∈A

Arbitrary A, B∈A, λ∈C

(i) (A*)* = A

(ii) (A+B)* = A*+B*

(iii) (λA)* =λ^-A*

(iiii) (AB)* = B*A*

<11 Linear topological space>

Number field     K

Linear space over K     X

When X satisfies next condition, X is called linear topological space.

(i) X is topological space

(ii) Next maps are continuous.

(x, y)∈X×X ↦ x+y∈X

(λ, x)∈K×X ↦λx∈X

Basis of neighborhoods of X’ zero element 0     V

When V⊂V is vertex set, X is called locally vertex.

<12 Adjoint space>

Norm space     X

Distance     d(x, y) = ||x-y|| (x, y∈X )

X is locally vertex linear topological space.

All of bounded linear functional over X    X*

Norm of f ∈X*      ||f||

X* is Banach space and is called adjoint space of X.

Adjoint space of X* is Banach space and is called the second adjoint space.

When X = X*, X is called reflective.

ⅡIndispensable theorems for proof

<1 Stone-Weierstrass Theorem>

Compact Hausdorff space     Ω

Subalgebra     A ⊂C(Ω)

When A ⊂C(Ω) satisfies next condition, A  is dense at C(Ω).

(i) A  separates points of Ω.

(ii) f∈A →　f^－∈A

(iii) 1∈A

Locally compact Hausdorff space        Ω

Subalgebra     A ⊂C₀(Ω)

When A ⊂C₀(Ω) satisfies next condition, A  is dense at C₀(Ω).

(i) A  separates points of Ω.

(ii) f∈A → f^－∈A

(iii) Arbitrary ω∈A ,  f∈A ,  f(ω) ≠0

<2 Norm algebra>

C* algebra     A

Arbitrary element of A     A

When A is normal, lim_n→∞||An||^1/n = ||A||

lim_n→∞||An||^1/n  is called spectrum radius of A. Notation is r(A).

[Note for norm algebra]

<2-1>

Number field     K = R or C

Linear space over K     X

Arbitrary elements of X     x, y

< x, y>∈K satisfies next 3 conditions is called inner product of x and y.

Arbitrary x, y, z∈X, λ∈K

(i) <x, x> ≧0,  <x, x> = 0 ⇔x = 0

(ii) <x, y> = 

(iii) <x, λy+z> = λ<x, y> + <x, z>

Linear space that has inner product is called inner space.

<2-2>

||x|| = <x, x>^1/2

Schwarz’s inequality

Inner space     X

|<x, y>|≦||x|| + ||y||

Equality consists of what x and y are linearly dependent.

||・|| defines norm over X by Schwarz’s inequality.

Linear space that has norm || ・|| is called norm space.

<2-3>

Norm space that satisfies next condition is called complete.

u_n∈X (n = 1, 2,…), lim_n, m→∞||u_n – u_m|| = 0

u∈X   lim_n→∞||u_n – u|| = 0

Complete norm space is called Banach space.

<2-4>

Topological space X that is Hausdorff space satisfies next condition is called normal.

Closed set of X     F, G

Open set of X     U, V

F⊂U, G⊂V, U∩V = Ø

<2-5>

When A  satisfies next condition, A  is norm algebra.

A  is norm space.

∀A, B∈A

||AB||≦||A|| ||B||

<2-6>

When A is complete norm algebra on || ・ ||, A is Banach algebra.

<2-7>

When A is Banach algebra that has involution * and || A*|| = ||A|| (∀A∈A),  A is Banach * algebra.

<2-8>

When A is Banach * algebra and ||A*A|| = ||A||²(∀A∈A) , A is C*algebra.

<3 Commutative Banach algebra>

Commutative Banach algebra     A

Arbitrary A∈A

Character X

|X(A)|≦r(A)≦||A||

[Note for commutative Banach algebra]  (   ) is referential section on this paper.

<3-1 Hilbert space>

Hilbert space     inner space that is complete on norm ||x||      Notation is H.

<3-2 Linear operator>

Norm space     V

Subset of V     D

Element of D     x

Map T : x → Tx∈V

The map is called operator.

D is called domain of T. Notation is D ( T ) or dom T.

Set A⊂D

Set TA     {Tx : x∈A}

TD is called range of T. Notation is R (T) or ran T.

α , β∈C,   x, y∈D ( T )

T(αx+βy) = αTx+βTy

T is called linear operator.

<3-3 Bounded linear operator>

Norm space     V

Subset of V     D

sup{||x|| ; x∈D} < ∞

D is called bounded.

Linear operator from norm space V to norm space V₁      T

D ( T ) = V

||Tx||≦γ (x∈V )  γ > 0

T is called bounded linear operator.

||T || := inf {γ : ||Tx||≦γ||x|| (x∈V)} = sup{||Tx|| ; x∈V, ||x||≦1} = sup{; x∈V,  x≠0}

||T || is called norm of T.

Hilbert space     H ,K

Bounded linear operator from H  to K     B (H, K )

B ( H ) : = B ( H, H )

Subset K ⊂H

Arbitrary x, y∈K, 0≦λ≦1

λx + (1-λ)y ∈K

K  is called vertex set.

<3-4 Homomorphism>

Algebra A  that has involution*       *algebra

Element of *algebra     A∈A

When A = A*, A is called self-adjoint.

When A *A= AA*, A is called normal.

When A A*= 1, A is called unitary.

Subset of A     B

B * := B*∈B

When B = B*, B is called self-adjoint set.

Subalgebra of A     B

When B is adjoint set, B is called *subalgebra.

Algebra     A, B

Linear map : A →B  satisfies next condition, π is called homomorphism.

π(AB) = π(A)π(B) (∀A, B∈A )

*algebra    A

When π(A*) = π(A)*, π is called *homomorphism.

When ker π := {A∈A ; π(A) =0} is {0},π is called faithful.

Faithful *homomorphism is called *isomorphism.

<3-5 Representation>

*homomorphism π from *algebra to B ( H ) is called representation over Hilbert space H of A .

<3-6 Character>

Homomorphism that is not always 0, from commutative algebra A  to C, is called character.

All of characters in commutative Banach algebra A  is called character space or spectrum space. Notation is Sp( A ).

<3-7 Gerfand representation>

Commutative Banach algebra     A

Homomorphism ∧: A →C(Sp(A))

∧is called Gerfand representation of commutative Banach algebra A.

<4 Gerfand-Naimark Theorem>

When A is commutative C* algebra, A  is equal distance *isomorphism to C(Sp(A)) by Gerfand representation.

[Note for Gerfand-Naimark Theorem]

<4-1 equal distance operator>

Operator     A∈B ( H )

Equal distance operator A     ||Ax|| = ||x|| (∀x∈H)

<4-2 Equal distance *isomorphism>

C* algebra      A

Homomorphism π

π(AB) = π(A)π(B) (∀A, B∈A )

*homomorphism   π(A*) = π(A)*

*isomorphism     { π(A) =0} = {0}

<5 Banach-Alaoglu theorem>

When X is norm space, (X*)₁ is weak * topology and compact.

[Note for Banach-Alaoglu theorem]

<5-1 Unit sphere>

Unit sphere X₁ := {x∈X ; ||x||≦1}

<5-2 Linear functional>

Linear space     V

Function that is valued by K     f (x)

When f (x) satisfies next condition, f is linear functional over V.

(i) f (x+y) = f (x) +f (y)   (x, y∈V)

(ii) f (αx) = αf (x)   (α∈K, x∈V)

<5-3 weak * topology>

All of Linear functionals from linear space X to K     L(X, K)

When X is norm space, X*⊂L(X, K).

Topology over X , σ(X, X*) is called weak topology over X.

Topology over X*, σ(X*, X) is called weak * topology over X*.

<6 *subalgebra of B ( H )>

When *subalgebra N of B ( H ) is identity operator I∈N , N ”= N is equivalent with τ^uw-compact.

[Note for *subalgebra of B ( H )]

<6-1 Identity operator>

Norm space     V

Arbitrary x∈V

Ix = x

I is called identity operator.

<6-2 Commutant>

Subset of C*algebra B (H)     A

Commutant of A     A ’

A ’ := {A∈B (H) ; [A, B] := AB – BA = 0, ∀B∈A }

Bicommutant of A     A ' ’’ := (A ’)’

A ⊂A ’’

<6-3 von Neumann algebra>

*subalgebra of C*algebra B (H)     A

When A  satisfies A ’’ = A  , A  is called von Neumann algebra.

<6-4 Ultra-weak topology>

Sequence of B ( H )     {A_α}

{A_α} is convergent to A∈B ( H )

Topology     τ

When α→∞, A_α →^τ A

Hilbert space     H

Arbitrary {x_n}, {y_n}⊂H

∑_n||x_n||² < ∞

∑_n||y_n||² < ∞

|∑_n<x_n, (A_α- A)y_n>| →0

A∈B ( H )

Notation is A_α →^uτ A

[ 7 Distance theorem]

For von Neumann algebra N over separable Hilbert space, N₁ can put distance on τ^s and τ^w topology.

[Note for distance theorem]

<7-1 Equipotent>

Sets     A, B

Map     f : A → B

All of B’s elements that are expressed by f(a) (a∈A)     Image(f)

a , a’∈A

When f(a) = f(a’) →a = a’, f is injective.

When Image(f) = B, f is surjective.

When f is injective and surjective, f is bijective.

When there exists bijective f from A to B, A and B are equipotent.

<7-2 Relation>

Sets     A, B

x∈A, y∈B

All of pairs <x, y> between x and y are set that is called product set between a and b.

Subset of product set A×B     R

R is called relation.

x∈A, y∈B, <x, y>∈R     Expression is xRy. 

When A =B, relation R is called binary relation over A.     

<7-3 Ordinal number>

Set     a

∀x∀y[x∈a∧y∈x→y∈a]

a is called transitive.

x, y∈a

x∈y is binary relation.

When relation < satisfies next condition, < is called total order in strict sense.

∀x∈A∀y∈A[x<y∨x=y∨y<x]

When a satisfies next condition, a is called ordinal number.

(i) a is transitive.

(ii) Binary relation ∈ over a is total order in strict sense.

<7-4 Cardinal number>

Ordinal number    α

α that is not equipotent to arbitrary β<α is called cardinal number.

<7-5 Cardinality>

Arbitrary set A is equipotent at least one ordinal number by well-ordering theorem and order isomorphism theorem.

The smallest ordial number that is equipotent each other is cardinal number that is called cardinality over set A. Notation is |A|.

When |A| is infinite cardinal number, A is called infinite set.

<7-6 Countable set>

Set that is equipotent to N     countable infinite set

Set of which cardinarity is natural number     finite set

Addition of countable infinite set and finite set is called countable set.

<7-7 Separable>

Norm space     V

When V has dense countable set, V is called separable.

<7-8 N₁>

von Neumann algebra     N   

A∈B ( H )

N₁ := {A∈N; ||A||≦1}

<7-9 τ^s and τ^w topology>

<7-9-1τ^s topology>

Hilbert space     H

A∈B ( H )

Sequence of B ( H )  {A_α}

{A_α} is convergent to A∈B ( H )

Topology     τ

When α→∞, A_α →^τ A

|| (A_α- A)x|| →0 ∀x∈H

Notation is A_α →^s A

<7-9-2 τ^w topology>

Hilbert space     H

A∈B ( H )

Sequence of B ( H )  {A_α}

{A_α} is convergent to A∈B ( H )

Topology     τ

When α→∞, A_α →^τ A

|<x, (A_α- A)y>| →0 ∀x, y∈H

Notation is A_α →^w A

<8 Countable elements>

von Neumann algebra N over separable Hilbert space is generated by countable elements.

<9 Only one real function>

For compact Hausdorff space Ω,C(Ω) that is generated by countable idempotent elements is generated by only on real function.

<9-1>

Set that is defined arithmetic・     S

Element of S     e

e satisfies a・e = e・a = a is called identity element.  

Identity element on addition is called zero element.

Ring’s element that is not zero element and satisfies a² = a is called idempotent element.

To be continued

Tokyo April 20, 2008

Sekinan Research Field of Language

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