The completion of a C*-algebra with a locally convex topology

There are examples of C*-algebras A that accept a locally convex*-topology t coarser than the given one, such that Ae[t] (the completion ofA with respect to t) is a GB*-algebra. The multiplication of A[t] may be ornot be jointly continuous. In the second case, Ae[t] may fail being a locallyconvex *-algebra, but it is a partial *-algebra. In both cases the structure andthe representation theory of Ae[t] are investigated. If A[t+]denotes the t-closureof the positive cone A+ of the given C*-algebra A, then the property A[t]+\cap (−A[t]+) = {0} is decisive for the existence of certain faithful *-representationsof the corresponding *-algebra Ae[t].