Wednesday 17 July 2013

Properly Infinite

Note 1
Properly Infinite  

On von Neumann algebra N, next are equivalent.
(i) N is properly infinite.
(ii) There exist {En : nN}P(N) and En~InEn = I.
(iii)There exist EP(N) and E~E~I.

<1 Objection Operator>
Hilbert space     H
Linear subspace of H     Subspace
Subspace that is closed by norm || || of H    Closed subspace
Arbitrary subspace of H         K
K: = {x; <xy> = 0, y K}     Orthogonal complement of K
Subspaces of H     KL
<xy> = 0 xK  yL     It is called that x and y are orthogonal each other. Notation is KL.
Direct sum KL : = {x+y ; xKyL}
= dist(xK) : = inf{||x-y|| ; yK}
d = ||x-z||
z : = PKx
PK is called objection operator from H to K.
von Neumann algebra     N
All of objection operators that belong to N     (N)
All of unitary operators that belong to N     U (N)

<2 Bounded operator>
Hilbert space      H, K
Subspace of H     D
Map    A
A(λx+μy) = λAx+μAyxyDλμC
A is called linear operator from H to K.
D     domain of A    Notation is dom A.
Set {Ax ; xD}     range of A    Notation is ran A.
dom A = H
Constant M>0
||Ax|M||x||  (xH)
A is called bounded operator from H to K
All of As     B(HK)
H = K
B(H:= B(H, H)
<xAy> = <A*xy>
A* is called adjoint operator of A.
A is called self-adjoint.
A*A = AA*
A is called normal operator.
A = A* = A2
A is called objection operator.
||Ax|| = ||x|| (xH)
A is called isometric operator.
A*A AA* I   I is identity operator.)
A is called unitary operator.
Ker A := {xH, Ax = 0}
A that is isometric over (Ker A) is called partial isometric operator.
von Neumann algebra     N
Commutant of N     N ‘
Center of N     Z := NN ‘      
Z = CI
N is called factor.
Central projection     E that belongs to Z    
All of central projections     P(Z)
Projection operator     EFP(N)
Partial isometric operator     WN
E ~ F1
Situation is expressed by  F.
 gives P(Npartial order relation.

<3 Comparison theorem>
For EFP(N), there exists PP(Z) , while EPFP and FPEP.

<4 Cardinality>
<4-1 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.     
<4-2 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.
<4-3 Cardinal number>
Ordinal number    α
α that is not equipotent to arbitrary β<α is called cardinal number.
<4-4 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.
<4-5 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.
<4-6 Zermelo’s well-ordering theorem>
If there exist Axiom of Choice, there exists well-ordering over arbitrary set.
<4-7 Order isomorphism theorem>
Arbitrary well-ordered set is order isomorphic to only one ordinal number.
<4-8 Axiom of choice>
xf Map(xx)∧∀y[yxy≠0 → f(y)y]]

To be continued
Tokyo May 1, 2008

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