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 -algebra with an identity. Consider the completed tensor product of *A* with itself with respect to the minimal or the maximal -tensor product norm. Assume that is a non-zero -homomorphism such that where is the identity map. Then is called a comultiplication on *A*. The pair can be thought of as a 'compact quantum semi-group'.

A left invariant Haar measure on the pair is a state on *A* such that for all . We show in this paper that a left invariant Haar measure exists if the set is dense in . It is not hard to see that, if also is dense, this Haar measure is unique and also right invariant in the sense that .

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).

**[1]**Edward G. Effros and Zhong-Jin Ruan,*Discrete quantum groups. I. The Haar measure*, Internat. J. Math.**5**(1994), no. 5, 681–723. MR**1297413**, 10.1142/S0129167X94000358**[2]**K. H. Hofmann,*Elements of compact semi-groups*, Merrill, Columbus, OH, 1966.**[3]**A. Van Daele,*The Haar measure on finite quantum groups*, Proc. Amer. Math. Soc.**125**(1997), no. 12, 3489–3500. MR**1415374**, 10.1090/S0002-9939-97-04037-9**[4]**-,*Quasi-discrete locally compact quantum groups*, preprint, K. U. Leuven, 1993.**[5]**-,*Discrete quantum groups*, preprint, K. U. Leuven, 1993.**[6]**S. L. Woronowicz,*Compact matrix pseudogroups*, Comm. Math. Phys.**111**(1987), no. 4, 613–665. MR**901157****[7]**-,*Compact quantum groups*, preprint, University of Warsaw, 1992.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1277138-0

Article copyright:
© Copyright 1995
American Mathematical Society