ar X iv : 0 90 5 . 12 96 v 1 [ m at h . O A ] 8 M ay 2 00 9 CONVOLUTION SEMIGROUPS OF STATES

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C0semigroups of maps which we characterise. On C∗-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.