Convolution semigroups of states

Details Convolution semigroups of states Convolution semigroups of states

2009-05-08

arxiv.org/abs/0905.1296v2

Mathematics

Operator Algebras

15 pages; accepted for publication in Mathematische Zeitschrift. v2 contains several minor corrections, main theorems remain u

Scientific paper

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 C_0-semigroups 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.

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