Algebry operatorów i ich zastosowania w fizyce

In 1954, R. Ellis proved the following remarkable result: Let S be a locally compact (Hausdorff) semi-topological semigroup, which is algebraically (i.e. as a set) a group. Then S is a topological group. A striking aspect of the theorem is that though the assumption "S is algebraically a group" is a purely "abstract algebraic" assumption, it somehow intervenes with the topology and as a result we get joint continuity of the product and continuity of the inverse. Recently, Matthew Daws introduced compact semi-topological quantum semigroup, as a tool to study weak almost periodicity of Hopf von Neumann algebras. We will first find a quantum analogue of the statement "algebraically a group" for these objects and then we will prove an Ellis theorem-type result for these objects. In particular, we will give a new proof of the classical Ellis theorem as well. Joint work with Colin Mrozinski.