The Haar measure on a compact quantum group

Abstract

Let A be a C ∗ {{\text {C}}^ \ast } -algebra with an identity. Consider the completed tensor product A ⊗ ¯ A A\bar \otimes A of A with itself with respect to the minimal or the maximal C ∗ {{\text {C}}^ \ast } -tensor product norm. Assume that Δ : A → A ⊗ ¯ A \Delta :A \to A\bar \otimes A is a non-zero ∗ ^ \ast -homomorphism such that ( Δ ⊗ ι ) Δ = ( ι ⊗ Δ ) Δ (\Delta \otimes \iota )\Delta = (\iota \otimes \Delta )\Delta where ι \iota is the identity map. Then Δ \Delta is called a comultiplication on A. The pair ( A , Δ ) (A,\Delta ) can be thought of as a ’compact quantum semi-group’. A left invariant Haar measure on the pair ( A , Δ ) (A,\Delta ) is a state φ \varphi on A such that ( ι ⊗ φ ) Δ ( a ) = φ ( a ) 1 (\iota \otimes \varphi )\Delta (a) = \varphi (a)1 for all a ∈ A a \in A . We show in this paper that a left invariant Haar measure exists if the set Δ ( A ) ( A ⊗ 1 ) \Delta (A) (A \otimes 1) is dense in A ⊗ ¯ A A\bar \otimes A . It is not hard to see that, if also Δ ( A ) ( 1 ⊗ A ) \Delta (A) (1 \otimes A) is dense, this Haar measure is unique and also right invariant in the sense that ( φ ⊗ ι ) Δ ( a ) = φ ( a ) 1 (\varphi \otimes \iota )\Delta (a) = \varphi (a)1 . The existence of a Haar measure when these two sets are dense was first proved by Woronowicz under the extra assumption that A has a faithful state (in particular when A is separable).