The Haar measure on a compact quantum group
Author:
A. Van Daele
Journal:
Proc. Amer. Math. Soc. 123 (1995), 31253128
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 nonzero 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 semigroup'. 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).
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 E. G. Effros and Z.J. Ruan, Discrete quantum groups I. The Haar measure, preprint, UCLA, 1993. MR 1297413 (95j:46089)
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 K. H. Hofmann, Elements of compact semigroups, Merrill, Columbus, OH, 1966.
 [3]
 A. Van Daele, The Haar measure on finite quantum groups, preprint, K. U. Leuven, 1992. MR 1415374 (98b:16036)
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 , Quasidiscrete locally compact quantum groups, preprint, K. U. Leuven, 1993.
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 , Discrete quantum groups, preprint, K. U. Leuven, 1993.
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 S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613665. MR 901157 (88m:46079)
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 , Compact quantum groups, preprint, University of Warsaw, 1992.
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DOI:
http://dx.doi.org/10.1090/S00029939199512771380
PII:
S 00029939(1995)12771380
Article copyright:
© Copyright 1995 American Mathematical Society
