Note 1
Properly Infinite
[Theorem]
On von Neumann algebra N, next are equivalent.
(i) N is properly infinite.
(ii) There exist {En : n∈N}⊂P(N) and En~I, ∑nEn = 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 : = PKx
PK 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
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