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The Haar measure on a compact quantum group. (English) Zbl 0844.46032

Summary: Let \(A\) be a \(C^*\)-algebra with an identity. Consider the completed tensor product \(A\overline {\otimes} A\) of \(A\) with itself with respect to the minimal or the maximal \(C^*\)-tensor product norm. Assume that \(\Delta: A\to A\overline {\otimes} A\) is a non-zero *-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 \(A \overline {\otimes} A\). 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).

MSC:

46L05 General theory of \(C^*\)-algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
22C05 Compact groups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L55 Noncommutative dynamical systems
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