THE HAAR MEASURE ON A COMPACT QUANTUM GROUP

Let A be a C*-algebra with an identity. Consider the completed tensor product A $($) over bar$$ XA of A with itself with respect to the minimal or the maximal C*-tensor product norm. Assume that Delta: A --> A $($) over bar$$ XA is a non-zero *-homomorphism such that (Delta X l)Delta = (l X Delta)Delta where l is the identity map. Then Delta is called a comultiplication on A. The pair (A, Delta) can be thought of as a 'compact quantum semi-group'. A left invariant Haar measure on the pair (A,Delta) is a state phi on A such that (l X phi)Delta(a) = phi(a)1 for all a is an element of A. We show in this paper that a left invariant Haar measure exists if the set Delta(A)(A X 1) is dense in A $($) over bar$$ XA. It is not hard to see that, if also Delta(A)(1 X A) is dense, this Haar measure is unique and also right invariant in the sense that (phi X l)Delta(a) = phi(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).