Noncommutative Mathematics for Quantum Systems

50 Noncommutative Mathematics for Quantum Systems

Cu(G),cf. [Wor88, BMT01]. This fact can be deduced from the
properties of the right regular representation, see [Wor98], or also
[MvD98, Section 5], [Kus97, Theorem 7.9].
We give here a detailed self-contained proof that uses explicitly
the invariance of the Haar state.

Proof LetHbe the Hilbert space of the GNS representation of
Pol(G)associated to the Haar statehand letξdenotes the related
(normalized) cyclic vector. Then for anya ∈ Pol(G) we have
h(a) = 〈ξ,λ(a)ξ〉, whereλis the left regular representation. We
denote by‖·‖rthe norm inCr(G), that is‖a‖r =‖λ(a)‖, where
‖·‖denotes the operator norm.
In a similar way, we associate the Hilbert spaceHt, the GNS
representationρtonHtand the normalized cyclic vectorξtto each
stateφt(φt = Φ◦j 0 t,cf.Definition 1.5.3). We have φt(a) =〈ξt,
ρt(a)ξt〉fora∈Pol(G).
We define the operators

it : H3v→v⊗ξt∈H⊗Ht

πt : H⊗Ht 3 v⊗w→〈ξt,w〉Htv∈H

Et : B(H⊗Ht) 3 X→πt◦X◦it∈B(H).

Since for eacht, it is an isometry and πt is contractive, Et is
contractive too:‖Et(X)‖=‖πt◦X◦it‖≤‖X‖.
Next we define

U:λ(Pol(G))ξ⊗ρt(Pol(G))ξt→H⊗Ht

λ(a)ξ⊗ρt(b)ξt7→λ(a( 1 ))ξ⊗ρt(a( 2 )b)ξt

and we check that it is an isometry with adjoint given by

U∗(λ(a)ξ⊗ρt(b)ξt) =λ(a( 1 ))ξ⊗ρt(S(a( 2 ))b)ξt.

Indeed, using the invariance of the Haar measure, we show thatU
is isometric

‖U(λ(a)ξ⊗ρt(b)ξt)‖^2 =‖λ(a( 1 ))ξ⊗ρt(a( 2 )b)ξt‖^2

=h(a∗( 1 )a( 1 ))φt(b∗a∗( 2 )a( 2 )b) = (h⊗φbt)(a∗( 1 )a( 1 )⊗a∗( 2 )a( 2 ))

= (h?φbt)(a∗a)=h(a∗a)φbt( 1 )=h(a∗a)φt(b∗b)=‖λ(a)ξ⊗ρt(b)ξt‖^2 ,