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Journal Article

# The Haar Measure on a Compact Quantum Group

A. Van Daele
Proceedings of the American Mathematical Society
Vol. 123, No. 10 (Oct., 1995), pp. 3125-3128
DOI: 10.2307/2160670
Stable URL: http://www.jstor.org/stable/2160670
Page Count: 4

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Topics: Haar measures, Preprints

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Let A be a C*-algebra with an identity. Consider the completed tensor product $A \overset\otimes A$ of A with itself with respect to the minimal or the maximal C*-tensor product norm. Assume that $\Delta: A \rightarrow A \overset\otimes A$ is a non-zero *-homomorphism such that (Δ ⊗ ı)Δ = (ı ⊗ Δ)Δ where ı is the identity map. Then Δ is called a comultiplication on A. The pair (A, Δ) can be thought of as a `compact quantum semi-group'. A left invariant Haar measure on the pair (A, Δ) is a state φ on A such that (ı ⊗ φ)Δ(a) = φ(a) 1 for all a ∈ A. We show in this paper that a left invariant Haar measure exists if the set Δ(A)(A ⊗ 1) is dense in $A \overset\otimes A$. It is not hard to see that, if also Δ(A) (1 ⊗ A) is dense, this Haar measure is unique and also right invariant in the sense that (φ ⊗ ı)Δ(a) = φ(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).