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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Convolution semigroups of states.
AU - Lindsay, Martin
AU - Skalski, Adam G.
N1 - 15 pages. Preprint, 24 June 2009. Published Online First™, 3 November 2009. The original publication is available at www.springerlink.com
PY - 2011/2
Y1 - 2011/2
N2 - 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 a 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.
AB - 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 a 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.
KW - Convolution
KW - quantum group
KW - C-bialgebra
KW - disrete semigroup
KW - quantum L\'evy process.
U2 - 10.1007/s00209-009-0621-9
DO - 10.1007/s00209-009-0621-9
M3 - Journal article
VL - 267
SP - 325
EP - 339
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 1432-1823
IS - 1-2
ER -