The Haar Measure on a Compact Quantum Group

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