46 Noncommutative Mathematics for Quantum Systems
functional onA. Here,M(B)denotes the set of multipliers onBof
a∗-algebraB, and by well definedness we mean that∆(a)(b⊗ 1 )
and∆(a)( 1 ⊗b)should belong toA⊗Afor alla,b∈A(viewed as
a subalgebra of M(A⊗A)). We refer the reader to [VD98] for
details.
If(A,∆)is a compact quantum group, then(Pol(G),∆|A)is an
algebraic quantum group (of compact type) and the Haar state is a
faithful left and right integral.
Fora∈Pol(G)we can defineha∈Pol(G)′by the formula
ha(b) =h(ab) forb∈Pol(G),wherehis the Haar state, and we denote byAˆthe space of linear
functionals onA=Pol(G)of the formhafora∈Pol(G).
The set Aˆ becomes an associative ∗-algebra with the
convolution of functionals as the multiplication:λ?μ = (λ⊗μ)
◦∆, and the involutionλ∗(x) =λ(S(x)∗)(λ,μ ∈Aˆ). The Hopf
structure is given as follows: the coproduct∆ˆ is the dual of the
product on Pol(G), the antipodeSˆis the dual toSand the counitεˆ
is the evaluation in 1. In particular, we haveSˆ(λ)(x) =λ(Sx)for
λ∈Aˆ,x∈Pol(G)and if∆ˆ(λ)∈A⊗ˆ Aˆthen
∆ˆ(λ)(x⊗y) =λ( 1 )(x)⊗λ( 2 )(y) =λ(xy), x,y∈Pol(G).The pairGˆ= (Aˆ,∆ˆ)is an algebraic quantum group, called thedual
ofG.
The linear map that associates toa∈Pol(G)the functionalha∈
Aˆis called theFourier transformand its value on an elementais also
denoted byaˆ. Let us note that, due to the faithfulness of the Haar
stateh,Aˆseparates the points of Pol(G).
Woronowicz characters and modular automorphism group
Fora∈A,λ∈A′we define
λ?a = (id⊗λ)∆(a),a?λ = (λ⊗id)∆(a).Ifa∈Pol(G)andλ∈Pol(G)′, thenλ?a,a?λ∈Pol(G).
For a compact quantum groupAwith the dense∗-Hopf algebra
Pol(G), there exists a unique family (fz)z∈C of linear
multiplicative functionals on Pol(G), calledWoronowicz characters
(cf.[Wor98, Theorem 1.4]), such that