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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The Haar measure on a compact quantum group


Author: A. Van Daele
Journal: Proc. Amer. Math. Soc. 123 (1995), 3125-3128
MSC: Primary 46L30; Secondary 46L60, 81R50
MathSciNet review: 1277138
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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).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277138-0
PII: S 0002-9939(1995)1277138-0
Article copyright: © Copyright 1995 American Mathematical Society