In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group. Definitions Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems. Definition (weight). Let be a C*-algebra, and let denote the set of positive elements of . A weight on is a function such that for all , and for all and . Some notation for weights. Let be a weight on a C*-algebra . We use the following notation: , which is called the set of all positive -integrable elements of . , which is called the set of all -square-integrable elements of . , which is called the set of all -integrable elements of . Types of weights. Let be a weight on a C*-algebra . We say that is faithful if and only if for each non-zero . We say that is lower semi-continuous if and only if the set is a closed subset of for every . We say that is densely defined if and only if is a dense subset of , or equivalently, if and only if either or is a dense subset of . We say that is proper if and only if it is non-zero, lower semi-continuous and densely defined. Definition (one-parameter group). Let be a C*-algebra. A one-parameter group on is a family of *-automorphisms of that satisfies for all . We say that is norm-continuous if and only if for every , the mapping defined by is continuous (surely this should be called strongly continuous?). Definition (analytic extension of a one-parameter group). Given a norm-continuous one-parameter group on a C*-algebra , we are going to define an analytic extension of . For each , let , which is a horizontal strip in the complex plane. We call a function norm-regular if and only if the following conditions hold: It