von Neumann Algebra 3 Note 1 Properly Infinite / 1 May 2008

von Neumann Algebra 3

Note 1

Properly Infinite  

TANAKA Akio

[Theorem]

On von Neumann algebra N, next are equivalent.

(i) N is properly infinite.

(ii) There exist {E_n : n∈N}⊂P(N) and E_n~I, ∑_nE_n = I.

(iii)There exist E∈P(N) and E~E^⊥~I.

[Explanation]

<1 Objection Operator>

<1-1>

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∈H ; <x, y> = 0, ∀y ∈K}     Orthogonal complement of K

Subspaces of H     K, L

<x, y> = 0 ∀x∈K  ∀y∈L     It is called that x and y are orthogonal each other. Notation is K⊥L.

Direct sum K⊕L : = {x+y ; x∈K, y∈L}

<1-2>

x∈H

d = dist(x, K) : = inf{||x-y|| ; y∈K}

z∈K

d = ||x-z||

z : = P_Kx

P_K is called objection operator from H to K.

<1-3>

von Neumann algebra     N

All of objection operators that belong to N     P (N)

All of unitary operators that belong to N     U (N)

<2 Bounded operator>

<2-1>

Hilbert space      H, K

Subspace of H     D

Map    A

A(λx+μy) = λAx+μAy, ∀x, y∈D, ∀λ, μ∈C

A is called linear operator from H to K.

D     domain of A    Notation is dom A.

Set {Ax ; x∈D}     range of A    Notation is ran A.

<2-2>

dom A = H

Constant M>0

||Ax|| ≦M||x||  (∀x∈H)

A is called bounded operator from H to K

All of As     B(H, K)

H = K

B(H) := B(H, H)

<2-3>

A∈B(H)

A*∈B(H)

<x, Ay> = <A*x, y>

A* is called adjoint operator of A.

A = A*

A is called self-adjoint.

A*A = AA*

A is called normal operator.

A = A* = A²

A is called objection operator.

||Ax|| = ||x|| (∀x∈H)

A is called isometric operator.

A*A = AA* = I   ( I is identity operator.)

A is called unitary operator.

Ker A := {x∈H, Ax = 0}

A that is isometric over (Ker A)^⊥ is called partial isometric operator.

<2-4>

von Neumann algebra     N

Commutant of N     N ‘

Center of N     Z := N∩N ‘      

Z = CI

N is called factor.

E∈P(N)

Central projection     E that belongs to Z    

All of central projections     P(Z)

<2-5>

Projection operator     E, F∈P(N)

Partial isometric operator     W∈N

F₁∈P(N)

F₁≦F

E ~ F₁

Situation is expressed by E ≺ F.

≺ gives P(N) partial order relation.

<3 Comparison theorem>

<3-1>

[Theorem]

For E, F∈P(N), there exists P∈P(Z) , while EP≺FP and FP^⊥≺EP^⊥.

<4 Cardinality>

<4-1 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.     

<4-2 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.

<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>

∀x∃f [ f ∈Map(x, ∪x)∧∀y[y∈x∧y≠0 → f(y)∈y]]

To be continued

Tokyo May 1, 2008

Sekinan Research Field of Language