is analytic on the interior of , i.e., for each in the interior of , the limit exists with respect to the norm topology on . It is norm-bounded on . It is norm-continuous on . Suppose now that , and let Define by . The function is uniquely determined (by the theory of complex-analytic functions), so is well-defined indeed. The family is then called the analytic extension of . Theorem 1. The set , called the set of analytic elements of , is a dense subset of . Definition (K.M.S. weight). Let be a C*-algebra and a weight on . We say that is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on if and only if is a proper weight on and there exists a norm-continuous one-parameter group on such that is invariant under , i.e., for all , and for every , we have . We denote by the multiplier algebra of . Theorem 2. If and are C*-algebras and is a non-degenerate *-homomorphism (i.e., is a dense subset of ), then we can uniquely extend to a *-homomorphism . Theorem 3. If is a state (i.e., a positive linear functional of norm ) on , then we can uniquely extend to a state on . Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is an ordered pair , where is a C*-algebra and is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions: The co-multiplication is co-associative, i.e., . The sets and are linearly dense subsets of . There exists a faithful K.M.S. weight on that is left-invariant, i.e., for all and . There exists a K.M.S. weight on that is right-invariant, i.e., for all and . From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight is automatically faithful. Therefore, the faithfulness of is a redundant condition and does not need to be postulated. Duality The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups. Alternative formulations The theory has an equivalent formulation in terms of von Neumann algebras. See also Locally compact space Locally compact field Locally compact group References Johan Kustermans & Stefaan