Stonesouth News

Thursday, 22 May 2025

The Days of Ideogram Ideogram from hieroglyph to LATEX Symbol List

The Days of Ideogram

Ideogram from hieroglyph to LATEX Symbol List

TANAKA Akio
Sekinan Library

Mark * in the text shows Reference number.

1. Ideogram
Ideogram has long history.
In the early paper I ever simply defined ideogram*0-1.
I also wrote relative papers on ideogram from the viewpoint of Chinese characters*3-1, 3-2, 3-3.
Especially Manuscript of Quantum Theory for Language 2003 is a starting root of my research on language universals*3-2.
The history of ideogram continues from Egyptian hieroglyph to Chinese character and now adds LATEX Symbol List*6-1.
The 21st century will necessarily become the century of ideogram.
The reasons are below.

Ideogram can contain rich information in small space.
Ideogram can show information at a glance.
Ideogram can sustain information stably in long time*1-7.
Ideogram has fertile capability to superb diversity of languages*8-5.

2. Set theory
Probably in summer 1977, I was designing for a paper on the sentence in language. The central part of the paper was connection between words or sentence, in which the logic of language would be shown using the set theory as a simplified model of true and false. I was strongly influenced from Kurt Godel, Gaishi Takeuchi and Bourbaki*1-1. But this design was abandoned for my lack of mathematical basis keeping on.

3. Qing dynasty's linguistics
In 1980s and 1990s I was lost in learning Chinese classical character theory, Xiaoxue, especially Qing dynasty for ideogram's fundamental property*2-1. At this period I did not write any paper on language. I solely read WANG Guowei*1-2, WANG Yinzhi, DUAN Yucai and Qing dynasty's linguists.

4. Time property in characters
In Autumn 2002 I got pneumonia and was hospitalized about 2 weeks, where I thought of 1970s' dream, writing clear description on language universals by mathematics. The theme was as hard as ever. So, at the bed I thought the basis of language from a side of Chinese character’s classical approach which had vast heritage till Qing dynasty. I directed my attention to the character's figure which had compound meanings containing time elements continuing from Yin dynasty's hieroglyphic characters left on bones and tortoise carapaces some 2400 years ago. I thought that Chinese characters had containing time and its structure could be written by geometric approach once I had abandoned for difficulty. After leaving hospital, I wrote a paper titled On Time Property Inherent in Characters*3-1.

5. Quantum elements
In 2003 staying at Hakuba, Nagano, I wrote a rial paper on language universals from the viewpoint of physics. This was remained manuscript till now. The title was Manuscript of Quantum Theory for Language*3-2 for the time being. Because in summer 2003 I wrote the first paper on language universals from physics titles as Quantum Theory for Language*3-3 which was read at a conference held in winter 2003 of Nara Japan.

6. Finite generative
Language probably begins from finite elements and generates.*3-6, 3-7 its world responding to outer world. I learned the concept of finite generative from von Neumann Algebra*4-1 at around 2006 and 2007.

7. Structure and dimension
Language probably has structure, which details are unknown now for me. But several phenomena hints me the existence of structure containing dimension. Its example is shown at the Crete's lie, on which I wrote a tiny paper titled as True-false problem of the Crete*3-8 in 2013.
More details on mathematical approach to dimension is shown at Algebraic Geometry Language*4-2.

8. Immanent time
Chinese character has immanent time in figure *3-1, 3-4. On the theme I wrote several papers between 2003 and 2005. Main papers related with the theme are seen in Early Paper and Early Paper 2*5-1, 5-2.
For the immanent time, I wrote a mathematical paper titles as Word as Infinite Loop Space*3-9.
In comparison with Chinese character, hieroglyph is very fantastic object on ideogram. For more details, hieroglyph is more complex for describing the concept using two functions, logogram expressing things' meaning and phonogram expressing words' pronunciation*0-2, 8-3.
Besides hieroglyph has also definitive element which is put at the end of the word. This element is very similar to Chinese character's meaning element which is called Pang 旁 in Chinese*8-3.
But even for learning hieroglyph's basis, I have not enough time now. I only like to see the relative events on hieroglyph at time to time . The exhibition held at Mori Arts Gallery in 2012, Tokyo titled as The British Museum ANCIENT EGYPTIAN BOOK OF THE DEAD JOURNEY THROUGH THE AFTERLIFE *8-2 was very fantastic, in which The Greenfield Papyrus was overwhelming.

9. Disposition, distance, flow and boundary
For thinking of language, I have been interested in four elements, disposition, distance flow and boundary*3-10,11,12, 13, which seem to become the basis of language.
especially at researching ideogram, disposition is fundamental to construct grammar.

10. Energy, dimension and distance
For more further research language , now I suppose at least three elements being based from mathematical description, which are energy, dimension and distance*3-14.

11. Mapping and category
Ideogram now has an important part in computer age. Image confirmation is very popular at internet banking system, which can contain vast information in one image, so password decoding is more difficult than phonogram or Arabic numerals.
Generally ideogram has vast information to the others using comparatively easy way.
This ideograms situation resembles mapping at category theory in mathematics. For this point I wrote rough sketch to apply category theory to language theory. Sketch name is Derived Category Language 2016*4-3.

Appendix
Sekinan Library's publicised papers are shown at SRFL Paper*7-1.
Theoretical development on language universals are shown at Genealogical Tree of Sekinan's Paper Sixth Edition*6-2.
Related words and concepts on language universals are shown at Appendix / Language between Sergej Karcevskij and string theory, one century's trace The 30th Anniversary of Sekinan Library Memorial Essay*6-1.
Relative themes on language universals at Sekinan Library are seen at Twitter Sekinan Library’s Moments*7-3.

References

Sekinan Library
Author
TANAKA Akio

0

40 years passed from I read WANG Guowei
For WITTGENSTEIN Ludwig Revised with Symplectic Language Theory and Floer Homology Language
Language between Sergej Karcevskij and string theory, one century's trace Preface
Karcevskij conjecture 1928 and Kawamata conjecture 2002

Outer sites and publication
8

This essay is unfinished.

Tokyo
20 April 2017
Sekinan Zoho

The Days of Decipherment

The Days of Decipherment

TANAKA Akio

On 20 July 2016 I went Tokyo National Museum, Ueno Park, Tokyo to see the exhibition A JOURNEY TO THE IMMORTALS: TREASURES OF ANCIENT GREECE, where I saw the linear A and B. It reminds me the youth days, so to say, the days of decipherment.

1960s -1970s is the age of decipherment in a sense. I was age 20 in 1967 and was learning language and literature at university. In 1958 John Chadwick's THE DECIPHERMENT OF LINEAR B was published from Cambridge University Press. At the preface of the book he wrote that the decipherment of linear B was told at Documents in Mycenaean Greek (Cambridge University Press, 1956) and Michael Ventris that deciphered the Linear B.

In the same age in Japan, Xixia wenzi (Xixia characters) in China was deciphered by NISHIDA Tatsuo (1928-2012) who wrote the analysis and grammar of Xixia characters through the paper Seikamoji no bunseki narabini Seikago bunpou no kenkyuu in 1962.
In almost the same time, Inca characters were studying to decipher. I frequently heard that Russian team developed largely.

In early 1970s I frequently went to Kanda, Tokyo where old bookshops were selling vast Oriental books at the Hakusan street and Yasukuni Street. I bought Chinese classics, especially linguistic classics written in the Qing dynasty and I read them almost every day containing the comparison with the western linguistic results. The Qing dynasty's heritage were DUAN Yucai, WANG Niansun, WANG Yingzhi and WANG Guowei and so forth. DUAN Yucai's Showenjezi zhu and WANG Guowei's Guantang jilin were the most important for me.

In France, 1960s was the days of Bourbaki that was one of the decipher of geometry by algebra, at least I thought so at that time. I sought and bought several Bourbaki's books at the old bookshops in Kanda, Tokyo,which is the largest old bookshop streets in Japan. But from my ability to mathematics Bourbaki was too much difficult to read on. From the days the long and winding road began to mathematics and its applicable study for language universals.

At the exhibition of ancient Greece I confirmed in particular that the stability of language was kept by letters and characters from the Linear A and Linear B.

Exhibition Catalogue numbers are the next.
The numbers 39 and 40 are Linear A. 41 and 77 are Linear B.

39. Clay juglet c. 1800 B.C. ~ c. 1700 B.C.
40. Clay bar c. 1700 B.C. ~ c. 1650 B.C.
41. Clay tablet c. 1375 B.C. ~ c. 1350 B.C.
77. Linear B bar and tablet c. the 13th century B.C.

For my part the stability has been one of the biggest themes on language phenomena since I was taught from CHINO Eiichi through the results of the Linguistic Circle of Prague, especially of Sergej Karcevskij.

The exhibition catalogue and Chadwick's book Japanese translated edition.

Reference

References 2

References 3

References 4

References 5

Tokyo
30 July 2016
Sekinan Library

Read more: http://srfl-paper.webnode.com/news/the-days-of-decipherment/

Sunday, 18 May 2025

von Neumann Algebra 2 Note Generation Theorem

von Neumann Algebra 2
Note
Generation Theorem

TANAKA Akio

[Main 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
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

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
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∈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 =
U is covering of Y. ∪U = ∪i∈I ⊃Y
Subfamily of U 　　V = (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) ≧0, = 0 ⇔x = 0
(ii) =
(iii) = λ +
Linear space that has inner product is called inner space.

<2-2>
||x|| = 1/2
Schwarz’s inequality
Inner space X
||≦||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| →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 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, ∈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 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
|| →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/

P.S. and Generation Theorem end here.
31 August 2020
Generation Theorem all text reprint.
T.A.

Friday, 21 March 2025

Letter to a friend. February 2024. On language as a given. Translated by Google. 2025

Friday, 21 March 2025

Letter to a friend. February 2024

Dear Sir,

Regarding Tsuneyuki KAWASAKI, I found a letter. I sent to a friend in my Gmail, so, now

I will send it to you.

The main thing I noticed was how amazing the teacher was at quoting the Man'yoshu.

As you also write in the letter, this was the turning point
for me to completely move away from history and return to my original language.

.......................................................

It's part of a letter.

The blue text remains the same.

.......................................................

After the lectures and seminars at Wako University were over and the students had gone home,

When I was talking face-to-face with the professor in the quiet lab,

It was an irreplaceable moment for me.

The conversation ranged from lectures and exercises to a wide range of topics.

The topics were wide-ranging, including the former First Higher School, his research at university, and his involvement with Mount Hiei after graduation.

In particular, he was deeply involved in the pioneering studies of Sanskrit, Tibetan, Pali, and other languages that led the way in philosophy and modern Buddhist studies .

It was a sign of deep respect.

At the end of those days, the teacher would sometimes say,

His words were, "I have just done what I was given."

I didn't understand what the teacher was trying to convey to me by using the word "given. "

Although it would generally be taken as a sign of humility, I didn't ask my teacher.

I knew that the teacher would rarely respond to my questions in more detail.

In the summer of 1982, the editing of a three-volume collection of historical writings to be published by the University of Tokyo Press was almost complete the proofreading stage was underway.

The author, the editors of the anthology, and the publishing company were all involved in this work.

Because I was involved in the chronology and bibliography, the publishing company asked me to also proofread the text of the entire anthology .

The university had already entered summer vacation and my regular night shift work had almost finished, so

From July to early September, I was dedicated to proofreading the text of all three volumes.

This was my first time doing such serious, responsible proofreading work.

I didn't know how to properly proofread, so I worked by going back and forth between the text and the typesetting many times.

And at that moment, the image of people who lived their lives wholeheartedly in history , something that had previously been completely invisible to me due to my lack of talent , emerged from each and every sentence of the professor's essay.

The teacher had quoted an exquisite poem in an exquisite place.

Although it is excluded from this collection, a careful reading of the first volume of the anthology, which deals with the relationship between ancient literature and history, reveals that

When just one poem by a poet who has only one or a few poems published in the Manyoshu was placed in the description of the professor's essay, the poet's entire life was revealed as if it were being surveyed .

More than any commentary on the poet or the waka poem, the deep and broad world that the poet had experienced emerged from .

The professor simply quoted that poem at a certain point in his essay .

The teacher truly " just did what he was told to do."

But through this encounter,

As a result, it has disappeared from history.

To me, my teacher's world was separate.

From now on, I will return to the world of language, which was my original subject .

The road from then on was long, long and winding.

His theory gave me a vague idea of my own state of being in this world . I too felt that
I had to do my own thing .

Language itself taken as a given.

February 11, 2024

TANAKA Akio

.......................................................

Uploaded

18 March 2025

Letter to a friend. February 2024

謹啓

川崎庸之について、友人に送った手紙がGmailに残っておりましたので、

ご送付申し上げます。

先生の万葉集の引用のすごさに接したことが中心ですが、

手紙の中にもありますが、
私が歴史から完全に離れ、もとの言語へと帰ってゆく契機となったことが記されています。

.......................................................

手紙の一部です。

青字も、もとのままです。

.......................................................

和光大学の講義と演習が終わり学生さんたちも帰ったあと、

静かになった研究室で、先生と相対でお話するときは、

私にはかけがえのないひと時でした。

会話は講義や演習から離れて、広い話題に及びました。

旧制一高のこと、大学での研究、卒業後に係わった比叡山のことなど多岐に渡りました。

殊に先生は哲学や近代仏教学を先導したサンスクリット、チベット、パーリ等の諸語を切り開いた諸先学に、

深い敬意を表していました。

そうした最後に先生が時折、述べられたのは、

「私は所与のことをしてきただけだから」ということばでした。

私には、先生が「所与のこと」ということばで、私に何を伝えようとしたかは、わかりませんでした。

一般には謙遜のことばとして受け取られますが、私は先生に尋ねることはしませんでした。

先生が私のそうした問いに、より詳しく話されることがほとんどないことを知っておりました。

1982年夏、東大出版会から刊行される全三巻の歴史著作選集の編集がほぼ終わり校正の段階になったとき、

著者、選集の各編集の先生、出版会がその作業に係わりましたが、

年譜と著作目録に私が係わりました関係から、併せて選集全体の本文の校正も行うことを出版会から頼まれました。

丁度大学はすでに夏休みとなり、私の夜の定時制の勤務もほぼ終了していましたので、

7月から9月初めまで、私はひたすら全三巻の本文校正に従事しました。

こうした本格的で責任のある校正作業は初めてでした。

本当の校正の方法は知りませんでしたので、私は本文と組版を何度も行き来しながら、作業にあたりました。

そしてこのとき、非才な私にはそれまで全く見えていなかった、歴史をひたむきに生きた人々の姿が、先生の論考の一文一文から浮かび上がりました。

先生は絶妙な位置に絶妙な一首を引用していたのです。

今回の論集からは除外されていますが、古代文学と歴史との係わりを扱った選集第一巻を精読すると、

万葉集に一首か数首しか掲載されていない歌人の、たった一首が先生の論考の叙述の中に置かれると、まるでその歌人の全生涯が通観されるかのように浮かび上がってきたのです。

その歌人や和歌のどのような解説よりも、その歌人の経てきた深く広い世界がたった31文字の歌そのものから、浮かび上がってきたのです。

先生はただその一首を論考のある位置に引用しただけなのです。

先生はまさしく「所与のことをしてきただけ」だったのです。

しかし私はこの邂逅を通して、

結果的に歴史から去って行くこととなりました。

私にとって、先生の世界は隔絶していました。

私はこれ以後、私本来の主題であった言語の世界に、帰って行くことになります。

途は以後、長く遠く折れ曲がっていましたが。

先生の論考が、私自身のこの世界でのありようを、かすかにながら示してくださったのです。
私もまた、私自身の所与のことを、なさねばならないと思ったのです。

言語そのものを所与のこととして。

2024年2月11日

.......................................................

謹白

TANAKA Akio

18 March 2025

SekinanTable

Thursday, 22 May 2025

The Days of Ideogram ​​​Ideogram from hieroglyph to LATEX Symbol List

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Sunday, 18 May 2025

von Neumann Algebra 2 Note Generation Theorem

Friday, 21 March 2025

Letter to a friend. February 2024. On language as a given. Translated by Google. 2025

Letter to a friend. February 2024

Letter to a friend. February 2024

The Days of Ideogram Ideogram from hieroglyph to LATEX Symbol List