Completely positive definite functions and Bochnerʼs theorem for locally compact quantum groups

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We prove two versions of Bochnerʼs theorem for locally compact quantum groups. First, every completely positive definite “function” on a locally compact quantum group G arises as a transform of a positive functional on the universal C⁎-algebra C0u(Gˆ) of the dual quantum group. Second, when 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(G), and when G is coamenable, the Banach ⁎-algebra L♯1(G) has a contractive bounded approximate identity.