Monday 5 January 2015

von Neumann Algebra Note 2 Tensor Product / 5 April 2008

von Neumann Algebra
Note 2
Tensor Product 

TANAKA Akio


1
Hilbert spaces     HK
Linear space     ⊕ K := { x  y ;  H K }    
x1  yx y2 = ( x1 + y1 )  ( x2 + y), λx  ) = λ λy
Inner product     <x1  y1 , x2  y2 > = <x1x2 > + <y1y2 >
H      direct sum Hilbert space
2
Hilbert spaces     HK
Direct product space     H × = { (uv) ; u  Hv  K }
Functional over H ×K     (u) = <ux> <v>
Linear space by functional    H K
= ∑n i=1λixiy H 
= ∑n j=1μjujvj H 
Inner product over H K     f> = ∑n i=1n j=1λiμj<xiuj><yivj>
Hilbert space with inner product is tensor product Hilbert space H K .


Tokyo April 5, 2008

Sekinan Research Field of Language

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