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**A construction for weights on \(C^*\)-algebras. Dual weights for \(C^*\)-crossed products.**
*(English)*
Zbl 0956.46046

Weights on \(C^\ast\)-algebras \(A\) (that is, additive positive functionals \(\varphi\) on the positive cone \(A^+\) that are allowed to take value \(+\infty\) and commute with multiplication by positive numbers) can be regarded as noncommutative, or quantum, analogues of infinite positive measures, and learning how to construct invariant weights on \(C^\ast\)-dynamical systems is important for the needs of the developing general theory of Haar measure on quantum groups. A weight \(\varphi\) is “densely defined” if the collection of elements at which \(\varphi\) assumes a finite value is everywhere dense in the \(C^\ast\)-algebra, and lower semi-continuous if all sets of the form \(\{x\in A: \varphi(x)\leq a\}\), \(a\geq 0\), are closed. The present paper gives a method for systematically constructing densely defined, lower semi-continuous weights on arbitrary \(C^\ast\)-algebras \(A\), beginning with a given non-degenerate representation of \(A\) on a Hilbert space \(\mathcal H\) and a net of bounded operators on \(\mathcal H\) satisfying certain properties. Further the authors generalize the theory of dual weights from von Neumann algebras to arbitrary \(C^\ast\)-algebras, and construct a canonical weight \(\widetilde\varphi\) on the crossed product \(A\times_\alpha G\) of a \(C^\ast\)-dynamical system \((A,G,\alpha)\), starting with an invariant lower semi-continuous weight \(\varphi\) on the original \(C^\ast\)-algebra \(A\). As an application of the construction, the authors give a \(C^\ast\)-algebraic construction of the Haar measure on the quantum \(E(2)\) group, previously obtained by S. Baaj [Astérisque. 323, 11-48 (1995; Zbl 0840.46036)] using methods of von Neumann algebra theory.

Reviewer: Vladimir Pestov (Wellington)

### MSC:

46L55 | Noncommutative dynamical systems |

46L51 | Noncommutative measure and integration |

20G42 | Quantum groups (quantized function algebras) and their representations |

### Keywords:

\(C^\ast\)-dynamical systems; weights; quantum \(E(2)\) group; Haar measure on quantum groups; crossed product### Citations:

Zbl 0840.46036
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\textit{J. Quaegebeur} and \textit{J. Verding}, Int. J. Math. 10, No. 1, 129--157 (1999; Zbl 0956.46046)

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### References:

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[3] | Combes F., Compos. Math. 23 pp 49– (1971) |

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