von Neumann Algebra Note 2 Tensor Product

von Neumann Algebra

Note 2

Tensor Product 

TANAKA Akio

1

Hilbert spaces     H, K

Linear space     H ⊕ K := { x ⊕ y ; x ∈ H, y ∈ K }    

x₁ ⊕ y₁ + x₂ ⊕ y₂ = ( x₁ + y₁ ) ⊕ ( x₂ + y₂ ), λ( x ⊕ y ) = λx ⊕ λy

Inner product     <x₁ ⊕ y1 , x₂ ⊕ y₂ > = <x₁, x2 > + <y₁, y₂ >

H ⊕ K     direct sum Hilbert space

2

Hilbert spaces     H, K

Direct product space     H ×K  = { (u, v) ; u ∈ H, v ∈ K }

Functional over H ×K    x ⊗ y (u, v ) = <u, x> <v, y >

Linear space by functional    H ⊙K

f = ∑ⁿ _i=1λ_ix_i⊗y_i ∈ H ⊙K 

g = ∑ⁿ _j=1μ_ju_j⊗v_j∈ H ⊙K 

Inner product over H ⊙K     < f, g > = ∑ⁿ _i=1∑ⁿ _j=1λ^―_iμ_j<x_i, u_j><y_i, v_j>

Hilbert space with inner product is tensor product Hilbert space H ⊙K .

Tokyo April 5, 2008

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