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These lecture notes are intended as an introduction to the theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e. C-algebras and von Neumann algebras. The presentation revolves around the definition of a locally compact quantum group as given in [KuV00a] and [KuV03].

By is meant the analysis arising from the natural operator filtration of a symmetric Fock space over a Hilbert space of squareintegrable vector-valued functions on the positive half-line. Current texts on quantum stochastics are the monograph [Par], the lecture notes [Mey], the St. Flour lectures [Bia], and the Grenoble lectures [Hud]. Excellent background together with a wealth of examples may be found in these, each of which has its own emphasis. The point of view of these notes is closest to [Bia], as far as the basic construction of quantum stochastic integrals goes. Beyond that, particular emphasis is given to .

Throughout these lectures B(H) will denote the von Neumann algebra of all bounded operators on a Hilbert space H. All our Hilbert spaces will be complex with an inner product 〈.,.〉 which is anti-linear in the first variable. Usually we restrict ourselves to separable Hilbert spaces.