Sekinan Table: August 2020

Letter to WPM. Maria Pires’ Schumann KINDERSZENEN 19 June 2020. P.S. added 9 August 2020 Generation Theorem Reprint 31 August 2020

Letter to WPM

Dear WPM,

It rains gently all day in Tokyo. Now duling the rainy season,
probably till July.
I listened to the CD of Maria Pires’ Schumann KINDERSZENEN, OP. 15.
Delicate and accurate.
Once in life, I should like to write such a fine paper, so delicate and accurate.

Kind regards,

T. A.

Tokyo
19 June 2020
Geometrization Language

Hydrangea is the June’s flower during the rainy season in Japan.

P.S.

I have never written a paper like Pires’ piano.
But several paper are very dear for me by various reasons.
One of the dearest papers is Generation Theorem written in 2008.
The paper has a memory of my age 20s, 1970s.
Those days I had a fresh dream, standing at the entrance of research as a youth, probably visiting to all the youth who hope to get a ticket to a respectable researcher.

Dream was making all the meanings of natural language from the one and only Empty set.
In those days I had devoted myself to Kurt Godel and his following researcher TAKEUCHI Gaishi.
Godel’s Incompleteness Theorem shows me the perfect meaning of Incompleteness of natural language.
So I started a very tiny one step to the grandeur to construct natural language generated from one and only empty set.

I learnt Bourbaki’s series bought at Kanda Tokyo, where the books seeking for maybe certainly got to you if you were roaming over shops till the narrow streets and remaining the power going up the rattling stairs.

But my ability towards the aim was very low and limited. And overlooking mathematics in those days, applied math using to the different fields probably was not enough arranged for the beginners like me.

Visiting the making a fresh start in my life to math was in 1990s end, age already nearly 50. Happily contemporary math level was fully spread and easy to enter for me.
I read math books day after day, especially of algebraic geometry, which was the most familiar for me and seemed to be applied to my study.
And at last my dream had come true at a tiny paper entitled Generation Theorem in 2008.

Cordially again,

T.A.
Tokyo
9 August 2020
Sekinan Geometry

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 {An}.
An = *An
Spectrum deconstruction       An = ∫1-1  λdEλ(n)
C*algebra that is generated by set { Eλ(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
τs topologyⅡ7-9
τw 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     <Ai; i∈I>, Ai∈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.
 
Compact>
3-1
Set     X
Subset of X     Y
Family of X’s subset that is not empty set     U = <Ui; i∈I>
U is covering of Y.     ∪U = ∪i∈I ⊃Y
Subfamily of U 　　V = <Ui; 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, limn→∞||An||1/n = ||A||
limn→∞||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.
un∈X (n = 1, 2,…), limn, m→∞||un – um|| = 0
u∈X   limn→∞||un – 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.
 
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 Gelfand representation of commutative Banach algebra A.
 
<4 Gelfand-Naimark Theorem>
When A is commutative C* algebra, A  is equal distance *isomorphism to C(Sp(A)) by Gelfand representation.
 
[Note for Gelfand-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 τuw-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 )     {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
Hilbert space     H
Arbitrary {xn}, {yn}⊂H
∑n||xn||2 < ∞
∑n||yn||2 < ∞
|∑n<xn, (Aα- A)yn>| →0
A∈B ( H )
Notation is Aα →uτ A
 
[ 7 Distance theorem]
For von Neumann algebra N over separable Hilbert space, N1 can put distance on τs and τw 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 ordinal 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 τs and τw topology>
<7-9-1τs topology>
Hilbert space     H
A∈B ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
|| (Aα- A)x|| →0 ∀x∈H
Notation is Aα →s A
<7-9-2 τw topology>
Hilbert space     H
A∈B ( H )
Sequence of B ( H )  {Aα}
{Aα} is convergent to A∈B ( H )
Topology     τ
When α→∞, Aα →τ A
|<x, (Aα- A)y>| →0 ∀x, y∈H
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.
 
<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
www.sekinan.org
 
Read more: https://srfl-paper.webnode.com/news/von-neumann-algebra-2-note-generation-theorem/