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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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