Monday 24 June 2013

von Neumann Algebra 2 Generation Theorem

Generation Theorem  

[Main Theorem]
<Generation theorem>
Commutative von Neumann Algebra is generated by only one self-adjoint operator.
[Proof outline]
is generated by countable {An}.
A= *An
Spectrum deconstruction       An = 1-1  λdEλ(n)
C*algebra that is generated by set { Eλ(n) ; λQ[-1, 1], nN}     A
A’’ = N
is commutative.
Existence of compact Hausdorff space Ω = Sp(A  )
A   C(Ω)
Element corresponded with fC(Ω)     AA
N is generated by A.

[Index of Terms]
|| . ||2-2
<x, y>2-1
adjoint space12
axiom of infinity1-8
axiom of power set1-4
axiom of regularity1-10
axiom of separation1-6
axiom of sum1-5
B ( H )3-3
Banach algebra2-6
Banach space2-3
Banach* algebra2-6
Banach-Alaoglu theorem5
basis of neighbor hoods4
binary relation7-2
bounded linear operator3-3
bounded linear operator, B ( H )3-3
C* algebra2-8
cardinal number7-3
cardinality, |A|7-5
character space (spectrum space), Sp( )3-6
closed set2-2
countable set7-6
countable infinite set7-6
( )3-2
dom( )3-2
domain, ( ), dom( )3-2
empty set1-9
equal distance operator4-1
Gerfand representation3-7
Gerfand-Naimark theorem4
Hausdorff space5
Hilbert space3-1
idempotent element9-1
identity element9-1
identity operator6-1
inner product2-1
inner space6
linear functional5-2
linear operator3-2
linear space6
linear topological space11
locally compact3-2
locally vertex11
norm algebra5
norm space2-2
open covering3-2
open set2-2
ordinal number7-3
product set7-2
r( )2
( )3-2
ran( )3-2
range, ( ), ran( )3-2
Schwarz’s inequality2-2
spectrum radius r( )2
Stone-Weierstrass theorem1
subtopological space2-3
system of neighborhoods4
τs topology7-9
τw topology7-9
the second adjoint space12
topological space2-2
total order in strict sense7-3
ultra-weak topology6-4
unit sphere5-1
vertex set3-3
von Neumann algebra6-3
weak topology5-3
weak * topology5-3
zero element9-1

[Explanation of indispensable theorems for main theorem]
<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     xy[zxzy]→x=y.
1-2 Set     ab
1-3 a is subset of b.    x[xaxb].Notation is abb-a = {xb ; xa} is complement of a.
1-4 Axiom of power set     xyz[zyzx]. Notation is (a).
1-5 Axiom of sum     xyz[zyw[zwwx]]. Notation is a.
1-6 Axiom of separation     xt= (t1, tn), formula φ(xt)     xtyz[zyzxφ(xt)].
1-7 Proposition of intersection     {xxb} = {xbxa} is set by axiom of separation. Notation is ab.
1-8 Axiom of infinity     x[0x∧∀y[yxy{y}x]].
1-9 Proposition of empty set     Existence of set a is permitted by axiom of infinity. {xaxx} is set and has not element. Notation of empty set is 0 or Ø.
1-10 Axiom of regularity     x[x≠0→y[yxyx=0].

<2 Topology>
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     <Ai; iI>, AiTiAi is belonged to T.       
(ii) AB T ABT
(iii) ØT, XT.
Set having T, (XT), is called topological space, abbreviated to X, being logically not confused.
Element of T is called open set.
Complement of Element of is called closed set.
Topological space     (XT)
Subset of X     Y
S ={AAT}
Subtopological space     (YS)   
Topological space is abbreviated to subspace.

<3 Compact>
Set     X
Subset of X     Y
Family of X’s subset that is not empty set     U = <UiiI>
U is covering of Y.     U = iI Y
Subfamily of U   V = <Uii> (JI)
V is subcovering of U.
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 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 BA, U is called basis of neighborhoods of point a.

<5 Hausdorff space>
Topological space that satisfies next condition is called Hausdorff space.
Distinct points of X     ab        
Neighborhood of a     U
Neighborhood of b     V
U= 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.

<8 Algebra>
C(Ω) and C0(Ω) satisfy the condition of algebra at product between points.
Subspace     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 AY≠Ø, Y is dense in X.

<10 Involution>
Involution over algebra A over C is map * that satisfies next condition.
Map * : AA  A*A
Arbitrary ABAλC
(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 satisfies next condition, X is called linear topological space.
(i) X is topological space
(ii) Next maps are continuous.
(xy)X×X  x+yX
(λx)K×X λxX
Basis of neighborhoods of X’ zero element 0     V
When Vis vertex set, X is called locally vertex.

<12 Adjoint space>
Norm space     X
Distance     d(xy) = ||x-y|| (xyX )
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.

Indispensable theorems for proof
<1 Stone-Weierstrass Theorem>
Compact Hausdorff space     Ω
Subalgebra     C(Ω)
When C(Ω) satisfies next condition,  is dense at C(Ω).
(i) A  separates points of Ω.
(ii) f fA
(iii) 1A
Locally compact Hausdorff space        Ω
Subalgebra     A C0(Ω)
When A C0(Ω) satisfies next condition,  is dense at C0(Ω).
(i) A  separates points of Ω.
(ii) f→ fA
(iii) Arbitrary ω,  f,  f(ω) ≠0

<2 Norm algebra>
C* algebra     A
Arbitrary element of A     A
When A is normal, limn→∞||An||1/n = ||A||
limn→∞||An||1/n  is called spectrum radius of A. Notation is r(A).

[Note for norm algebra]
Number field     R or C
Linear space over K     X
Arbitrary elements of X     xy
xy>K satisfies next 3 conditions is called inner product of x and y.
Arbitrary xyzX, λK
(i) <xx0,  <xx> = 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.

||x|| = <xx>1/2
Schwarz’s inequality
Inner space     X
|<xy>|||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.

Norm space that satisfies next condition is called complete.
un(n = 1, 2,…), limnm→∞||un – um|| = 0
uX   limn→∞||un – u|| = 0
Complete norm space is called Banach space.

Topological space that is Hausdorff space satisfies next condition is called normal.
Closed set of X     FG
Open set of X     UV

When A  satisfies next condition, A  is norm algebra.
A  is norm space.
||AB||||A|| ||B||

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

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

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

<3 Commutative Banach algebra>
Commutative Banach algebra     A
Arbitrary AA
Character X

[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  TxV
The map is called operator.
D is called domain of T. Notation is D ) or dom T.
Set AD
Set TA     {Tx : xA}
TD is called range of T. Notation is (T) or ran T.
α , βC,   x, yD ( )
T(αx+βy) = αTx+βTy
T is called linear operator.

<3-3 Bounded linear operator>
Norm space     V
Subset of V     D
sup{||x|| ; xD} < ∞
D is called bounded.
Linear operator from norm space V to norm space V1      T
D ) = V
||Tx||γ (xV )  γ > 0
is called bounded linear operator.
||T || := inf {γ : ||Tx||γ||x|| (xV)} = sup{||Tx|| ; xV, ||x||1} = sup{xV,  x0}
||T || is called norm of T.
Hilbert space     H ,K
Bounded linear operator from H  to K     (H)
B ( H ) : = B ( H)
Subset K H
Arbitrary xyK, 0λ1
λx + (1-λ)y K
K  is called vertex set.

<3-4 Homomorphism>
Algebra A  that has involution*       *algebra
Element of *algebra     AA
When A = A*, A is called self-adjoint.
When A *AAA*, 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, is called *subalgebra.
Algebra     AB
Linear map : A B  satisfies next condition, π is called homomorphism.
π(AB) = π(A)π(B) (ABA )
*algebra    A
When π(A*) = π(A)*, π is called *homomorphism.
When ker π := {AA ; π(A) =0} is {0},π is called faithful.
Faithful *homomorphism is called *isomorphism.

<3-5 Representation>
*homomorphism π from *algebra to ) 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  is called character space or spectrum space. Notation is Sp( A ).

<3-7 Gerfand representation>
Commutative Banach algebra     A
Homomorphism 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     AB ( H )
Equal distance operator A     ||Ax|| = ||x|| (xH)

<4-2 Equal distance *isomorphism>
C* algebra      A
Homomorphism π
π(AB) = π(A)π(B) (ABA )
*homomorphism   π(A*) = π(A)*
*isomorphism     { π(A) =0} = {0}

<5 Banach-Alaoglu theorem>
When X is norm space, (X*)is weak * topology and compact.

[Note for Banach-Alaoglu theorem]
<5-1 Unit sphere>
Unit sphere X:= {xX ; ||x||1}

<5-2 Linear functional>
Linear space     V
Function that is valued by K     f (x)
When (x) satisfies next condition, f is linear functional over V.
(i) f (x+y) = (x) +(y)   (xyV)
(ii) (αx) = αf (x)   (αKxV)

<5-3 weak * topology>
All of Linear functionals from linear space X to K     L(XK)
When X is norm space, X*L(XK).
Topology over X , σ(XX*) 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 of B ( H ) is identity operator IN ”= N is equivalent with τuw-compact.

[Note for *subalgebra of B ( H )]
<6-1 Identity operator>
Norm space     V
Arbitrary xV
Ix x
I is called identity operator.

<6-2 Commutant>
Subset of C*algebra (H)     A
Commutant of A     A ’
A ’ := {A(H) ; [AB] := AB – BA = 0, B}
Bicommutant of A     A ' ’’ := (A ’)’

<6-3 von Neumann algebra>
*subalgebra of C*algebra (H)     A
When A  satisfies ’’ =  ,  is called von Neumann algebra.

<6-4 Ultra-weak topology>
Sequence of B ( H )     {Aα}
{Aα} is convergent to AB ( H )
Topology     τ
When α→∞, Aα τ A
Hilbert space     H
Arbitrary {xn}, {yn}H
n||xn||2 < 
n||yn||2 < 
|n<xn, (AαA)yn>| 0
AB ( H )
Notation is Aα  A

[ 7 Distance theorem]
For von Neumann algebra N over separable Hilbert space, N1 can put distance on τs and τtopology.

[Note for distance theorem]
<7-1 Equipotent>
Sets     AB
Map     f : A  B
All of B’s elements that are expressed by f(a) (aA)     Image(f)
a , aA
When f(a) = f(a’a = a’, f is injective.
When Image(f) = Bf is surjective.
When f is injective and surjective, f is bijective.
When there exists bijective f from A to Band B are equipotent.

<7-2 Relation>
Sets     AB
All of pairs <xy> between x and y are set that is called product set between a and b.
Subset of product set A×B     R
is called relation.
xAyB, <xy>R     Expression is xRy. 
When A =B, relation R is called binary relation over A.     

<7-3 Ordinal number>
Set     a
a is called transitive.
xy is binary relation.
When relation < satisfies next condition, < is called total order in strict sense.
When 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 has dense countable set, V is called separable.

<7-8 N1>
von Neumann algebra     N   
AB ( H )
N:= {AN; ||A||1}

<7-9 τs and τtopology>
<7-9-1τs topology>
Hilbert space     H
AB ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to AB ( H )
Topology     τ
When α→∞, Aα τ A
|| (AαA)x|| xH
Notation is Aα s A
<7-9-2 τtopology>
Hilbert space     H
AB ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to AB ( H )
Topology     τ
When α→∞, Aα τ A
|<x, (AαA)y>| xyH
Notation is Aα 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.

Set that is defined arithmetic     S
Element of S     e
e satisfies aea = a is called identity element.  
Identity element on addition is called zero element.
Ring’s element that is not zero element and satisfies ais called idempotent element.

To be continued
Tokyo April 20, 2008

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