Operator topology

A topology on the space of continuous linear mappings from one topological vector space into another topological vector space , converting the space into a topological vector space. Let be a locally convex space and let be a family of bounded subsets of such that the linear hull of the union of the sets of this family is dense in . Let be a basis of neighbourhoods of zero in . The family

where runs through and through , is a basis of neighbourhoods of zero for a unique topology that is invariant with respect to translation, which is an operator topology and which converts the space into a locally convex space; this topology is called the -topology on .

Examples. I) Let be locally convex spaces. 1) Let be the family of all finite subsets in ; the corresponding -topology (on ) is called the topology of simple (or pointwise) convergence. 2) Let be the family of all convex balanced compact subsets of ; the corresponding topology is called the topology of convex balanced compact convergence. 3) Let be the family of all pre-compact subsets of ; the corresponding -topology is called the topology of pre-compact convergence. 4) Let be the family of all bounded subsets; the corresponding topology is called the topology of bounded convergence.

II) If are Banach spaces considered simultaneously in the weak or strong (norm) topology, then the corresponding spaces coincide algebraically; the corresponding topologies of simple convergence are called the weak or strong operator topologies on . The strong operator topology majorizes the weak operator topology; both are compatible with the duality between and the space of functionals on of the form , where , , .

III) Let be Hilbert spaces and let be countable direct sums of the Hilbert spaces , respectively, where , for all integer ; let be the imbedding of the space into defined by the condition that for any operator the restriction of the operator to the subspace maps into and coincides on with the operator . Then the complete pre-image in of the weak (strong) operator topology on is called the ultra-weak (correspondingly, ultra-strong) operator topology on . The ultra-weak (ultra-strong) topology majorizes the weak (strong) operator topology. A symmetric subalgebra of the algebra of all bounded linear operators on a Hilbert space , containing the identity operator, coincides with the set of all operators from that commute with each operator from that commutes with all operators from , if and only if is closed in the weak (or strong, or ultra-weak, or ultra-strong) operator topology, i.e. is a von Neumann algebra.

References

[1]	H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[2]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988)
[3]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[4]	S. Sakai, "-algebras and -algebras" , Springer (1971)