Uniform topologies and strong operator topologies on polynomial algebras and on the algebra of CCR

Abstract

In the paper uniform topologies and strong operator topologies on the free polynomial algebra in n Hermitian indeterminants, on the polynomial algebra in n commuting Hermitian indeterminants and on the ∗-algebra generated by the CCR (finite number of degrees of freedom) are investigated. It is proved that the strongest locally convex topology on these algebras is a uniform topology and a strong operator topology. For the polynomial algebra in one variable it is shown that on each algebraical realization as an Op∗-algebra by an unbounded operator, the strongest locally convex topology coincides with the uniform topology. If in addition the realization is closed, then also the strong operator topology is equal to the strongest locally convex topology.