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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.

MSC:

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

Citations:

Zbl 0840.46036
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References:

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[4] DOI: 10.1112/plms/s3-66.3.619 · Zbl 0792.46040
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[7] DOI: 10.1112/plms/s3-50.1.131 · Zbl 0577.46063
[8] DOI: 10.1007/BF02100032 · Zbl 0743.46080
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