Ultrastrong topology  From Wikipedia, the free encyclopedia Jump
to: navigation, search  In functional analysis, the ultrastrong
topology, or σ-strong topology, or strongest topology on the set
B(H) of bounded operators on a Hilbert space is the topology
defined by the family of seminorms  p_\omega(x) = \omega(x^{*}
x)^{1/2}  for positive elements \omega of the predual L_{*}(H)
that consists of trace class operators. [1]:68  It was
introduced by von Neumann in 1936. [2]  Relation with the strong
(operator) topology[edit]  The ultrastrong topology is similar
to the strong (operator) topology. For example, on any norm-
bounded set the strong operator and ultrastrong topologies are
the same. The ultrastrong topology is stronger than the strong
operator topology.  One problem with the strong operator
topology is that the dual of B(H) with the strong operator
topology is "too small". The ultrastrong topology fixes this
problem: the dual is the full predual B*(H) of all trace class
operators. In general the ultrastr