sigma-Strong Operator Topology

Given a Hilbert space , the -strong operator topology is the topology on the algebra of bounded operators from to itself defined as follows: A sequence of operators in converges to an operator -strongly if and only if for all compact operators . Here, denotes the algebra of bounded operators from to itself.

The -strong topology is sometimes referred to as the ultrastrong topology due to it being "stronger" than the strong topology.

One can prove that the -strong topology on is generated by the collection of seminorms , as above, where here, .

The -strong topology is important for a number of reasons, not the least of which is its application to the study of von Neumann algebras. What's more, the notion of the -strong topology is merely one in a larger hierarchical class of operator topologies on which includes the -weak topology, the -strong* topology, etc.; this hierarchy is the focus of considerable study in its own right.