After all I came back to the very dream that I had embraced since the high school days. It was a fundamental ask on language related with mathematics and physics. The root of language would be able to describe by mathematics and physics. In my mind language is always put at the centre of the pursuit that was what anyone can clearly understand. Description by mathematics, but physics why (Note 1)?. Physics treats with substance that constructs the world in which I had desire to let language enter. It started at Substantiality of language (References 2. 2). It was my dream and probably will be so, hereafter.
Letter to O. again Bridge from Old China's Linguistics (Xiaoxue) to Linguistic Circle of Prague
20/05/2019 18:54
Letter to O. again Bridge from Old China's Linguistics (Xiaoxue) to Linguistic Circle of Prague
From TANAKA, again send a letter. Thanks a lot for your kind letter although you are in every day busy work.
I had thought again sending a letter to you but I could not, for year-end thorough cleaning of my little house.
The fact is I uploaded the letter to you on my Blogger SRFL News, removing your private part. This blog is still now ranked in the third position. My blog is written by English so the readers are almost foreigners. But this blog is written in Japanese as same as the real letter to you. Your sincere thinking of old books may give the readers with agreement, who love the old books like you.
In your letter, you wrote that you would read the books comfortably not selling the old books of your work. I have also felt the same feeling so I bought Development of the modern poems written by ANDO Tsuguo who was the fine teacher of literature at the university and I once went to his home thinking to hear his intimate voice on poem and literature. He was one of the most typical poet and critic in the present age of Japan.
And now I would want to read KIYOOKA Takayuki's collected poems, that was ever read in my youth.
In autumn 1970 I was the senior of university, when the next spring I would graduate, I met with the thick French binding KIYOOKA's collected poems at the well drop-in bookshop of Hachioji, Tokyo. At that time I could not yet find my whole life object of learning.
In his book, I read a long poem titled "at the campus of university" in which he wrote that he meets with the whole life object of learning as he meets with his beloved person in his life.
The next spring 1971 I graduated the university, when I could not find any object at the campus while I ruminated KIYOOKA's stanza.
What I really met with my whole life object was already entered in the 21 century at my age 56 in 2003 after much trial and error. It was language universals that was ever taught from CHINO Eiichi who was the real life time teacher of mine and the successor of the Linguistic Circle of Prague, who spent 7 year study at Prague.
Sergej Karcevskij gave me the great hint of approach to language universals through the famous paper titled "Du Dualisme asymétrique du signe linguistique" 1929.
In 2003 I wrote a paper titled " Quantum Theory for Language" after some trial papers, " On Time Property Inherent in Characters", " Manuscript of Quantum Theory for Language" . In 2004 I wrote two attached papers, " Distance Theory" and "Reversion Theory".
After the suggestion of CHINO, I have tried the very hard road of writing style using mathematical base assisted from many contemporary mathematician works, especially at the field of algebraic geometry.
The paper " Quantum Theory for Language" was selected one of the 4 oral presentations of the linguistic field of a international symposium focused on the Silk Road.
On Karcevskij and CHINO I wrote a tiny paper, " Prague in 1920" which was dedicated the two as a token of my cordially gratitude.
Dear O. I am very happy to meet with you so you gave me the true value of books and the precious relationship between books and human being. Your work and suggestion made me recollect the important people and events on my life.
From Author: This translation was automatically done. So there are indefinite expressions in the essay. I humbly beg your pardon. The original text can be read at the upper Light-blue URL.
1. 1963 Theoretical physics
In April 1963, I entered Tokyo Metropolitan Tachikawa High School in Tachikawa, Tokyo. The high school is located at the south exit of the former Japanese National Railways Tachikawa Station, a 10-minute walk away, and was an old school founded in 1901 as Prefectural Nichu . Tachikawa Station is about 30 km west of the city centre, and along with Hachioji Station, it is the central station in the Tama area.
On the day of the entrance ceremony, I remember that the cherry blossoms in full bloom were beautifully scattered outside the school gate, just like yesterday . On the left side of the school gate, there was a gatehouse that had been used for a long time, and in front of the central entrance, on the right side, there was a temporary hand-planted pine tree by Emperor Taisho. There were some old buildings left in the school, some of which were still in use while I was in school.
The gatehouse is an old-fashioned building with a round roof on the left side of the school gate . I was often there. When I joined the soccer club, I said .
There were 400 students in the first grade, which seems to be a remnant of the old junior high school, which was only for boys, but there were 300 boys and 100 girls. I was in the men's class for the first and third years, and the men and women's class only in the second year. The bare- footedclassroom was . After school shortly afterward entrance, only our class boys had cleaned the classrooms and corridors. Then a female employee flew in from the office so we didn't have to clean this school. I confirmed , and most of them probably never cleaned .
Kaneko, my best friend when I was three years old , told me with a grin that he had never cleaned it. The most fun part of my three years was when I was in the third year, even though it must have been difficult for me to take the university entrance exam .Kaneko and Kaneko, who were in the same class for the first time in three years, spent the year in the noisy seats at the back of the classroom, closest to the corridor side. I can't remember the details of why this happened now, but maybe the quiet and good seats naturally fill up, and Kaneko and I, who didn't care much about that, ended up sitting in the rest of the seats. It might have been. However, thanks to that, it was a free and enjoyable year, just like Yaji Kita Dochu.
There's a square trash can behind us, and quite a few boxes, especially as the one in front throws the used math calculator did not go to it, but it was scattered all over. If it gets worse , no one will get rid of it, so I had. Kaneko, of course, never went.
Growing up in a small town near the north-western tip of Tokyo with a population of about 20,000, where hills, forests and fields still occupied about 70 to 80%, I often went to the station by bicycle in the morning. On rainy days, I had to walk, and the hem of my feet got wet on the road in the woods, so I sometimes wore boots. Then, in the classroom , Kaneko often said . " To be sure, I rarely . That was me. Kaneko was good at mathematics. He surely solved the problem I was having and told me. Perhaps English was the only one I was almost equal to. He was always at the desk on the right side, sticking together, and said, "Tanaka, do you have a 1000-yen note?" He replied , "I don't have it," and fooled me, "Well, I can solve it with the watermark."
We, Group G, were not clearly limited, but since the classes were Mathematics III, Physics / Chemistry, and Japanese History / World Kaneko aspired to chemistry, and I aspired to physics . It was theoretical physics. I had a strong admiration for theoretical physics since the fall of the year. The one I longed for most was Shinichiro Tomonaga. I can't confirm exactly when I read the teacher's "Stay-German Diary", but while staying in Germany, he was aiming to build a new theoretical physics with many prominent physicists. I was fascinated by. In the fall of 1965, the morning after Dr Tomonaga's Nobel Prize in Physics was reported, a moment before the start of work , we at Group G talked about the topic around Kaneko and the blackboard behind my seat . I still remember one of the class happily saying .
Much later, when I was focusing on the time inherent in letters, I felt the need to sort out the ultra-multi-time theory that was one of the core of Professor Tomonaga's Nobel Prize , and Nishijima, known for his German theory. With reference to Professor Kazuhiko's dissertation , when I was able to understand the rigorous path of super-multi-time theory from Schrodinger to Dirac to Tomonaga as mathematics, I felt relieved as if I had solved my long homework since high school. put in memory. After publishing this article in a magazine, Professor Nishijima, I learned in the newspaper that he had died shortly thereafter. I think it was close to the teacher's last writing. I became one of the people who felt the grace of Professor Nishijima. The title of his dissertation at this time was " Dirac and Quantum Field Theory," Mathematical Science, September 2007, pp. 15-20, Science , Inc. 2007 .
It was natural for me to strongly aspire to the Faculty of Science at Kyoto University. However, I was probably the only one in Group G who wanted to go to Kyoto . Kyoto was a tough choice for me, but it didn't change while in March 1966, I headed . At that time, I was carrying a book of Dante's Divine Comedy translated by Soichi Nogami of the Chikuma Shobo version, which I read little by little . As a result of the examination, Kaneko passed the University of Tokyo II successfully, but I failed the 20 students of the Faculty of Science, Kyoto University . At that time, Kyoto University told me the scoring results of the exam via high school, so I had them sent to me after a while . As a result, it was a little more, but now, with my ability at that time, it would have been difficult to proceed to theoretical physics, even if it was included. However, I also learned that in terms of scoring , I was able to pass in some other faculties. This is the end of my high school life . I visited Kaneko's house several times after graduating from high school, who invited me to say, "It's not much different from your place ." On the slope at the foot of the Tama Hills, the red flowers of Oleander were beautiful in the summer . After finishing his college, he became his desired chemistry researcher. After a long Winding Road, I'll finally find the subject and how to pursue it, and I'll be asked, "Are you still doing that he Has already passed away. He was in his mid-50s. I lost .
References
TOMONAGA'S SUPER MULTI-TIME THEORY RESTING ELBPWS NEARLY PRAYER
Dear Bier, your study is very difficult to my poor understanding on social science. But I believe that you have contributed towards peace in the world by your piled-up work. I would like to learn the situation of the tight days little by little. T. A.
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.
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 a book seeking for, maybe invariably gets 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.
von Neumann Algebra 2NoteGeneration 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 = *AnSpectrum deconstruction An = ∫1-1λdEλ(n)C*algebra that is generated by set { Eλ(n) ; λ∈Q∩[-1, 1], n∈N} AA’’ = NA is commutative.I∈AExistence of compact Hausdorff space Ω = Sp(A )A = C(Ω)Element corresponded with f∈C(Ω) A∈AN 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 Seta, 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 ofpower 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 separationx, 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 XYS ={A∩Y ; A∈T}
Subtopological space (Y, S)
Topological space is abbreviated to subspace.
Compact>
3-1
Set X
Subset of XY
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 UV = <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 Xa
Subset of X A
Open set Ba∈B⊂AA is called neighborhood of a.
All of point a’s neighborhoods is called system of neighborhoods.
System of neighborhoods of point aV(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 aU
Neighborhood of bVU∩V = Ø
<6 Linear space>
Compact Hausdorff space ΩLinear 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∈Sab∈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 KX
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<1Stone-Weierstrass Theorem>Compact Hausdorff space ΩSubalgebra 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∈ALocally compact Hausdorff space ΩSubalgebra 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<2Norm algebra>C* algebra AArbitrary element of A AWhen A is normal, limn→∞||An||1/n= ||A||limn→∞||An||1/nis called spectrum radius of A. Notation is r(A).[Note for norm algebra]<2-1>
Number field K = R or C
Linear space over KX
Arbitrary elements of Xx, 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>1/2Schwarz’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 XF, G
Open set of XU, VF⊂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 AArbitrary A∈ACharacter 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 VD
Element of Dx
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+βTyT is called linear operator.
<3-3 Bounded linear operator>
Norm space V
Subset of VD
sup{||x|| ; x∈D} < ∞
D is called bounded.
Linear operator from norm space V to norm space V1 TD ( 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 ∈KK 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 AA*= 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 forGelfand-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 sphereX1 := {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∈VIx = 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, Bx∈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 RR 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∈ax∈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 αα 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 Ncountable 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 τsand 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 See 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 continuedTokyo April 20, 2008Sekinan Research Field of Languagewww.sekinan.org
Read more: https://srfl-paper.webnode.com/news/von-neumann-algebra-2-note-generation-theorem/
P.S. and Generation Theorem end here. 31 August 2020 Generation Theorem all text reprint. T.A.
