## 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.

### MSC:

 46L55 Noncommutative dynamical systems 46L51 Noncommutative measure and integration 20G42 Quantum groups (quantized function algebras) and their representations

Zbl 0840.46036
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### References:

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