von Neumann Algebra Note 1 Measure

von Neumann Algebra

Note 1

Measure 

TANAKA Akio

1

locally compact space     Ω

compact subset of Ω     K

continuous complex valued function space     K (Ω)

linear functional over K (Ω)     μ

positive number     λ

f ∈K (Ω) 

| μ (f) | ≦ λ sup_ω∈Ω | f (ω)|

μ is measure over Ω.

2

sequence of points     {x_n}, {y_n} ⊂ H

∑_n || x_n ||² < ∞

∑_n || y_n ||² < ∞

3

Hilbert space     H

bounded operator from H  to K     B (H, K )

B ( H ) : = B ( H, H )

sub-algebra that is self-adjoint set     *sub-algebra

*sub-algebra of B ( H ) that contains identity operator I and is closed by ultra weak topology (τ^uw)

*sub-algebra of B ( H )     N

linear functional A∈N ↦ ∑_n < x_n, Ay_n> is τ^uw-continuous.

Tokyo April 3, 2008

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