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Completely positive definite functions and Bochner’s theorem for locally compact quantum groups. (English) Zbl 1320.46056
Authors’ abstract: We prove two versions of Bochner’s theorem for locally compact quantum groups. First, every completely positive definite ‘function’ on a locally compact quantum group $$\mathbb G$$ arises as a transform of a positive functional on the universal $$C^*$$-algebra $$C^u_0(\hat{\mathbb G})$$ of the dual quantum group. Second, when $$\mathbb G$$ is coamenable, complete positive definiteness may be replaced with the weaker notion of positive definiteness, which models the classical notion. A counterexample is given to show that the latter result is not true in general. To prove these results, we show two auxiliary results of independent interest: products are linearly dense in $$L^1_\sharp(\mathbb G)$$, and when $$\mathbb G$$ is coamenable, the Banach $$*$$-algebra $$L^1_\sharp(\mathbb G)$$ has a contractive bounded approximate identity.

##### MSC:
 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 43A35 Positive definite functions on groups, semigroups, etc. 43A20 $$L^1$$-algebras on groups, semigroups, etc. 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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