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|�V7-5

|| . ||�U2-2

||x||�U2-2

<x, y>�U2-1

*algebra�U3-4

*homomorphism�U3-4

*isomorphism�U3-4

*subalgebra�U3-4

adjoint space�T12

algebra�T8

axiom of infinity�T1-8

axiom of power set�T1-4

axiom of regularity�T1-10

axiom of separation�T1-6

axiom of sum�T1-5

B ( H )�U3-3

Banach algebra�U2-6

Banach space�U2-3

Banach* algebra�U2-6

Banach-Alaoglu theorem�U5

basis of neighbor hoods�T4

bicommutant�U6-2

bijective�U7-1

binary relation�U7-2

bounded�U3-3

bounded linear operator�U3-3

bounded linear operator, B ( H )�U3-3

C* algebra�U2-8

cardinal number�U7-3

cardinality, |A|�U7-5

character�U3-6

character space (spectrum space), Sp( )�U3-6

closed set�T2-2

commutant�U6-2

compact�T3-2

complement�T1-3

complete�U2-3

countable set�U7-6

countable infinite set�U7-6

covering�T3-1

commutant�U6-2

D ( )�U3-2

dense�T9

dom( )�U3-2

domain, D ( ), dom( )�U3-2

empty set�T1-9

equal distance operator�U4-1

equipotent�V7-1

faithful�U3-4

Gerfand representation�U3-7

Gerfand-Naimark theorem�U4

H�U3-1

Hausdorff space�T5

Hilbert space�U3-1

homomorphism�U3-4

idempotent element�U9-1

identity element�U9-1

identity operator�U6-1

injective�V7-1

inner product�U2-1

inner space�T6

involution*�T10

linear functional�U5-2

linear operator�U3-2

linear space�T6

linear topological space�T11

locally compact�T3-2

locally vertex�T11

N�V3-8

N₁�V3-8

neighborhood�T4

norm�U2-2

norm�U3-3

norm algebra�U5

norm space�U2-2

normal�U2-4

normal�U3-4

open covering�T3-2

open set�T2-2

operator�U3-2

ordinal number�U7-3

product�T8

product set�U7-2

r( )�U2

R ( )�U3-2

ran( )�U3-2

range, R ( ), ran( )�U3-2

reflective�T12

relation�V7-2

representation�U3-5

ring�T7

Schwarz’s inequality�U2-2

self-adjoint�U3-4

separable�U7-7

set�T7

spectrum radius r( )�U2

Stone-Weierstrass theorem�U1

subalgebra�T8

subcovering�T3-1

subring�T7

subset�T1-3

subspace�T2-3

subtopological space�T2-3

surjective�V7-1

system of neighborhoods�T4

τ^s topology�U7-9

τ^w topology�U7-9

the second adjoint space�T12

topological space�T2-2

topology�T2-1

total order in strict sense�U7-3

ultra-weak topology�V6-4

unit sphere�U5-1

unitary�U3-4

vertex set�U3-3

von Neumann algebra�U6-3

weak topology�U5-3

weak * topology�U5-3

zero element�U9-1

[Explanation of indispensable theorems for main theorem]

�TPreparation

<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.

�UIndispensable 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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