Noncommutative Mathematics for Quantum Systems

64 Noncommutative Mathematics for Quantum Systems

generated by X and Y, then W is unitary.

Proof Define W on simple tensors of bounded continuous
functions by

W f⊗g=g(Y)f(X)Ω

forf,g∈Cb(C). It follows from the monotone independence ofX
andYthat this defines an isomorphism, since

〈W f 1 ⊗g 1 ,W f 2 ⊗g 2 〉 = 〈Ω,f 1 (X)∗g 1 (Y)∗g 2 (Y)f 2 (X)Ω〉

= 〈Ω,f 1 (X)∗f 2 (X)Ω〉〈Ω,g 1 (Y)∗g 2 (Y)Ω〉

=

∫

f 1 (t)f 2 (t)dμ(t)

∫

g 1 (t)g 2 (t)dν(t).

SinceCb(C)⊗Cb(C)is dense inL^2 (C×C,μ⊗ν),Wextends to a
unique isomorphism onL^2 (C×C,μ⊗ν).
The relations

〈W f 1 ⊗g 1 ,h(X)W f 2 ⊗g 2 〉=〈Ω,f 1 (X)∗g 1 (Y)∗h(X)g 2 (Y)f 2 (X)Ω〉

=〈Ω,f 1 (X)∗

(

h(X)−h( 0 )

)

f 2 (X)Ω〉〈Ω,g 1 (Y)∗Ω〉〈Ω,g 2 (Y)Ω〉

+h( 0 )〈Ω,f 1 (X)∗g 1 (Y)∗g 2 (Y)f 2 (X)Ω〉

=〈Ω,g 2 (Y)Ω〉

〈

W f 1 ⊗g 1 ,W

(

(h−h( 0 ) 1

)

f 2 ⊗ 1

〉

+h( 0 )〈W f 1 ⊗g 1 ,W f 2 ⊗g 2 〉

=

〈

W f 1 ⊗g 1 ,W

(∫

g 2 (y)dν(y)(h−h( 0 ) 1 )f 2 ⊗ 1 +h( 0 )f 2 ⊗g 2

)〉

and

〈W f 1 ⊗g 1 ,h(Y)W f 2 ⊗g 2 〉 = 〈Ω,f 1 (X)∗g 1 (Y)∗h(Y)g 2 (Y)f 2 (X)Ω〉

= 〈W f 1 ⊗g 1 ,W f 2 ⊗(hg 2 )〉

show that we have the desired formulae for simple tensors of
functions f 1 ,f 2 ,g 1 ,g 2 ∈ Cb(C). The general case follows by
linearity and continuity. Remark 1.7.10(c) implies

W L^2 (C×C,μ⊗ν) =span{g(Y)f(X)Ω;f,g∈Cb(C)}

=alg{h(X),h(Y);h∈Cb(C)}Ω.

IfΩis cyclic, thenWis surjective and therefore unitary.