The Haar measure on a compact quantum group

Abstract:

Let A be a ${{\text {C}}^ \ast }$-algebra with an identity. Consider the completed tensor product $A\bar \otimes A$ of A with itself with respect to the minimal or the maximal ${{\text {C}}^ \ast }$-tensor product norm. Assume that $\Delta :A \to A\bar \otimes A$ is a non-zero $^ \ast$-homomorphism such that $(\Delta \otimes \iota )\Delta = (\iota \otimes \Delta )\Delta$ where $\iota$ 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 $\varphi$ on A such that $(\iota \otimes \varphi )\Delta (a) = \varphi (a)1$ for all $a \in A$. We show in this paper that a left invariant Haar measure exists if the set $\Delta (A) (A \otimes 1)$ is dense in $A\bar \otimes A$. It is not hard to see that, if also $\Delta (A) (1 \otimes A)$ is dense, this Haar measure is unique and also right invariant in the sense that $(\varphi \otimes \iota )\Delta (a) = \varphi (a)1$. 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).