Algebry operatorów i ich zastosowania w fizyce

Let G be a locally compact quantum group. In this talk we will describe the quantum counterpart of the center Z(G) of G. It will be shown that Z(G) is a strongly normal coideal of G and the explicit description of the quotient quantum group G/Z(G) will be given. In the second part of the talk we shall consider an action alpha of G on a von Neumann algebra. Although in general it is impossible to define ker alpha, we shall present the construction of the quantum counterpart of G/(ker alpha) together with the faithful quotient action of G/(ker alpha) on M. We shall show that for alpha being the inner action of G on the von Neumann algebra of the dual hatG, G/(ker alfa) equals G/Z(G). This is joint work with P. Sołtan.