This paper by P. Porcelli and E. A. Pedersen discusses the properties of rings of operators on a complex Hilbert space. They consider an abelian symmetric subring E of the ring of bounded operators on the Hilbert space, which is closed in the weak operator topology. The commutant of E, denoted E₁, is shown to have a cyclic vector ξ₀, which is normalized to have unit norm. Dixmier showed that E and E₁ are duals of certain Banach spaces. The authors show that every linear functional T on E is continuous in the weak operator topology, implying that continuity in any of the four topologies (weak, ultraweak, strong, ultrastrong) is equivalent. They also derive an integral representation for such T and use it in a theorem on centrally reducible positive functionals on E₁. The maximal ideal space of E is denoted M, and the Gel'fand transforms of elements of E are denoted by a, b, etc. The mapping from E to C(M) is an isometric isomorphism. Every bounded linear functional on E can be represented as an integral over M with respect to a complex Borel measure. Functionals of the form A → (Aξ, ξ) are of particular interest, and their corresponding measures are shown to be nonnegative and dominated by μ, the measure corresponding to ξ₀. Theorem 1 states that a linear functional T on E is ultrastrongly continuous if and only if it is weakly continuous, and if and only if there exists a function φ in L₁(M, μ) such that T(A) is the integral of a(m)φ(m) with respect to μ. The proof involves showing that such a functional is weakly continuous by constructing a sequence of functions bₙ that approximate φ and using properties of the Gel'fand transform. Theorem 2 states that a weakly continuous positive functional on E₁ is centrally reducible. The proof involves showing that the restriction of such a functional to E is weakly continuous and using the integral representation to find an operator B such that the functional can be expressed as T(AB).This paper by P. Porcelli and E. A. Pedersen discusses the properties of rings of operators on a complex Hilbert space. They consider an abelian symmetric subring E of the ring of bounded operators on the Hilbert space, which is closed in the weak operator topology. The commutant of E, denoted E₁, is shown to have a cyclic vector ξ₀, which is normalized to have unit norm. Dixmier showed that E and E₁ are duals of certain Banach spaces. The authors show that every linear functional T on E is continuous in the weak operator topology, implying that continuity in any of the four topologies (weak, ultraweak, strong, ultrastrong) is equivalent. They also derive an integral representation for such T and use it in a theorem on centrally reducible positive functionals on E₁. The maximal ideal space of E is denoted M, and the Gel'fand transforms of elements of E are denoted by a, b, etc. The mapping from E to C(M) is an isometric isomorphism. Every bounded linear functional on E can be represented as an integral over M with respect to a complex Borel measure. Functionals of the form A → (Aξ, ξ) are of particular interest, and their corresponding measures are shown to be nonnegative and dominated by μ, the measure corresponding to ξ₀. Theorem 1 states that a linear functional T on E is ultrastrongly continuous if and only if it is weakly continuous, and if and only if there exists a function φ in L₁(M, μ) such that T(A) is the integral of a(m)φ(m) with respect to μ. The proof involves showing that such a functional is weakly continuous by constructing a sequence of functions bₙ that approximate φ and using properties of the Gel'fand transform. Theorem 2 states that a weakly continuous positive functional on E₁ is centrally reducible. The proof involves showing that the restriction of such a functional to E is weakly continuous and using the integral representation to find an operator B such that the functional can be expressed as T(AB).